let M1, M2 be Matrix of REAL; :: thesis: ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = abs (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) = abs (M * (i,j)) ) implies M1 = M2 )
assume that
A5: len M1 = len M and
A6: width M1 = width M and
A7: for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = abs (M * (i,j)) and
A8: ( len M2 = len M & width M2 = width M ) and
A9: for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = abs (M * (i,j)) ; :: thesis: M1 = M2

now

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A10: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
A11: dom M1 = dom M by A5, FINSEQ_3:29;
hence M1 * (i,j) = abs (M * (i,j)) by A6, A7, A10
.= M2 * (i,j) by A6, A9, A10, A11 ;
:: thesis: verum

end;

hence M1 = M2 by A5, A6, A8, MATRIX_1:21; :: thesis: verum