let M1, M2 be Matrix of K; ( len M1 = len A & width M1 = width B & ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) & len M2 = len A & width M2 = width B & ( for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) implies M1 = M2 )
assume that
A5:
len M1 = len A
and
A6:
width M1 = width B
and
A7:
for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = (Line (A,i)) "*" (Col (B,j))
and
A8:
len M2 = len A
and
A9:
width M2 = width B
and
A10:
for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) = (Line (A,i)) "*" (Col (B,j))
; M1 = M2
dom M2 = dom M1
by A5, A8, FINSEQ_3:29;
then A11:
( Indices M1 = [:(dom M1),(Seg (width M1)):] & Indices M2 = [:(dom M1),(Seg (width M1)):] )
by A6, A9;
now for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)let i,
j be
Nat;
( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )assume A12:
[i,j] in Indices M1
;
M1 * (i,j) = M2 * (i,j)then
M1 * (
i,
j)
= (Line (A,i)) "*" (Col (B,j))
by A7;
hence
M1 * (
i,
j)
= M2 * (
i,
j)
by A10, A11, A12;
verum end;
hence
M1 = M2
by A5, A6, A8, A9, MATRIX_0:21; verum