let n, m be Nat; :: thesis: for D being non empty set
for M1, M2 being Matrix of n,m,D st ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ) holds
M1 = M2
let D be non empty set ; :: thesis: for M1, M2 being Matrix of n,m,D st ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ) holds
M1 = M2
let M1, M2 be Matrix of n,m,D; :: thesis: ( ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ) implies M1 = M2 )
A1: len M1 = n by Th25;
A2: len M2 = n by Th25;

A3: now :: thesis: width M1 = width M2

per cases ( n = 0 or n > 0 ) ;

suppose A4: n = 0 ; :: thesis: width M1 = width M2

then width M1 = 0 by A1, Def3;
hence width M1 = width M2 by A2, A4, Def3; :: thesis: verum

end;

suppose A5: n > 0 ; :: thesis: width M1 = width M2

then width M1 = m by A1, Th20;
hence width M1 = width M2 by A2, A5, Th20; :: thesis: verum

end;

end;

end;

assume for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ; :: thesis: M1 = M2
hence M1 = M2 by A1, A2, A3, Th21; :: thesis: verum