TS002E · theorem body

two_square_product_difference_forward

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

The nonnegative a*d-b*c branch gives an explicit natural Brahmagupta representation.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

forall a b c d m. a * d = b * c + m -> (a * a + b * b) * (c * c + d * d) = (a * c + b * d) * (a * c + b * d) + m * m

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall a b c d m. a * d = b * c + m -> (a * a + b * b) * (c * c + d * d) = (a * c + b * d) * (a * c + b * d) + m * m

Proof neighborhood

Direct theorem prerequisites

TS002C two_square_product_norm_expanded TS002B two_square_cross_products_equal TS002D two_square_balanced_difference_identity

Direct theorem dependents

TS002G two_square_product_explicit_witness

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

23 script commands · 5 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (3)
two square product norm expandedtwo square cross products equaltwo square balanced difference identity
01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro m
  6. L6
    intro hdifference
02Use earlier factsL7–10

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L7
    specialize two_square_product_norm_expanded a
  2. L8
    specialize two_square_product_norm_expanded b
  3. L9
    specialize two_square_product_norm_expanded c
  4. L10
    specialize two_square_product_norm_expanded d
03Calculate and transport equalitiesL11–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    rewrite two_square_product_norm_expanded
04Use earlier factsL12–21

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L12
    specialize two_square_balanced_difference_identity (a * c)
  2. L13
    specialize two_square_balanced_difference_identity (b * d)
  3. L14
    specialize two_square_balanced_difference_identity (a * d)
  4. L15
    specialize two_square_balanced_difference_identity (b * c)
  5. L16
    specialize two_square_balanced_difference_identity m
  6. L17
    apply two_square_balanced_difference_identity
  7. L18
    specialize two_square_cross_products_equal a
  8. L19
    specialize two_square_cross_products_equal b
  9. L20
    specialize two_square_cross_products_equal c
  10. L21
    specialize two_square_cross_products_equal d
05Use earlier factsL22–23

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L22
    exact two_square_cross_products_equal
  2. L23
    exact hdifference

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro m
  6. 0006intro hdifference
  7. 0007specialize two_square_product_norm_expanded a
  8. 0008specialize two_square_product_norm_expanded b
  9. 0009specialize two_square_product_norm_expanded c
  10. 0010specialize two_square_product_norm_expanded d
  11. 0011rewrite two_square_product_norm_expanded
  12. 0012specialize two_square_balanced_difference_identity (a * c)
  13. 0013specialize two_square_balanced_difference_identity (b * d)
  14. 0014specialize two_square_balanced_difference_identity (a * d)
  15. 0015specialize two_square_balanced_difference_identity (b * c)
  16. 0016specialize two_square_balanced_difference_identity m
  17. 0017apply two_square_balanced_difference_identity
  18. 0018specialize two_square_cross_products_equal a
  19. 0019specialize two_square_cross_products_equal b
  20. 0020specialize two_square_cross_products_equal c
  21. 0021specialize two_square_cross_products_equal d
  22. 0022exact two_square_cross_products_equal
  23. 0023exact hdifference