let D be non empty set ; :: thesis: for M1, M2 being Matrix of D st M1 @ = M2 @ & len M1 = len M2 holds
M1 = M2
let M1, M2 be Matrix of D; :: thesis: ( M1 @ = M2 @ & len M1 = len M2 implies M1 = M2 )
assume that
A1: M1 @ = M2 @ and
A2: len M1 = len M2 ; :: thesis: M1 = M2
len (M1 @) = width M1 by Def6;
then A3: width M1 = width M2 by A1, Def6;
A4: Indices M2 = [:(dom M2),(Seg (width M2)):] ;
for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A5: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
dom M1 = Seg (len M2) by A2, FINSEQ_1:def 3
.= dom M2 by FINSEQ_1:def 3 ;
then (M2 @) * (j,i) = M2 * (i,j) by A3, A4, A5, Def6;
hence M1 * (i,j) = M2 * (i,j) by A1, A5, Def6; :: thesis: verum

end;

hence M1 = M2 by A2, A3, Th21; :: thesis: verum