let n, m be Nat; :: thesis: for D being non empty set
for M1, M2 being Matrix of n,m,D holds Indices M1 = Indices M2
let D be non empty set ; :: thesis: for M1, M2 being Matrix of n,m,D holds Indices M1 = Indices M2
let M1, M2 be Matrix of n,m,D; :: thesis: Indices M1 = Indices M2
A1: len M1 = n by Def2;
A2: len M2 = n by Def2;

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;

dom M1 = Seg n by A1, FINSEQ_1:def 3;
hence Indices M1 = Indices M2 by A2, A3, FINSEQ_1:def 3; :: thesis: verum