let M1, M2 be Matrix of REAL; ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = |.(M * (i,j)).| ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = |.(M * (i,j)).| ) implies M1 = M2 )
assume that
A6:
len M1 = len M
and
A7:
width M1 = width M
and
A8:
for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = |.(M * (i,j)).|
and
A9:
( len M2 = len M & width M2 = width M )
and
A10:
for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = |.(M * (i,j)).|
; M1 = M2
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 A11:
[i,j] in Indices M1
;
M1 * (i,j) = M2 * (i,j)A12:
dom M1 = dom M
by A6, FINSEQ_3:29;
hence M1 * (
i,
j) =
|.(M * (i,j)).|
by A7, A8, A11
.=
M2 * (
i,
j)
by A7, A10, A11, A12
;
verum end;
hence
M1 = M2
by A6, A7, A9, MATRIX_0:21; verum