let S1, S2 be Matrix of n,m,D; :: thesis: ( ( for i, j being Nat st [i,j] in Indices S1 holds
S1 * i,j = M * (nt . i),(mt . j) ) & ( for i, j being Nat st [i,j] in Indices S2 holds
S2 * i,j = M * (nt . i),(mt . j) ) implies S1 = S2 )
assume that
A2: for i, j being Nat st [i,j] in Indices S1 holds
S1 * i,j = M * (nt . i),(mt . j) and
A3: for i, j being Nat st [i,j] in Indices S2 holds
S2 * i,j = M * (nt . i),(mt . j) ; :: thesis: S1 = S2

now

let i, j be Nat; :: thesis: ( [i,j] in Indices S1 implies S1 * i,j = S2 * i,j )
assume A4: [i,j] in Indices S1 ; :: thesis: S1 * i,j = S2 * i,j
A5: [i,j] in Indices S2 by A4, MATRIX_1:27;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def 13;
S1 * i,j = M * (nt . i9),(mt . j9) by A2, A4;
hence S1 * i,j = S2 * i,j by A3, A5; :: thesis: verum

end;

hence S1 = S2 by MATRIX_1:28; :: thesis: verum