let M1, M2 be Matrix of COMPLEX; ( len M1 = len x & width M1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') ) & len M2 = len x & width M2 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') ) implies M1 = M2 )
assume that
A8:
len M1 = len x
and
A9:
width M1 = len y
and
A10:
for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *')
and
A11:
len M2 = len x
and
A12:
width M2 = len y
and
A13:
for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = ((x . i) * (M * (i,j))) * ((y . 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 A14:
[i,j] in Indices M1
;
M1 * (i,j) = M2 * (i,j)A15:
dom M1 = dom M
by A1, A8, FINSEQ_3:29;
hence M1 * (
i,
j) =
((x . i) * (M * (i,j))) * ((y . j) *')
by A2, A9, A10, A14
.=
M2 * (
i,
j)
by A2, A9, A13, A14, A15
;
verum end;
hence
M1 = M2
by A8, A9, A11, A12, MATRIX_0:21; verum