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 & len M2 = n ) by Def3;
then A2: ( dom M1 = Seg n & dom M2 = Seg n ) by FINSEQ_1:def 3;
A3: 0 <= n by NAT_1:2;

now

per cases ( n = 0 or n > 0 ) by A3, XXREAL_0:1;

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

then ( width M1 = 0 & width M2 = 0 ) by A1, Def4;
hence width M1 = width M2 ; :: thesis: verum

end;

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

then ( width M1 = m & width M2 = m ) by A1, Th20;
hence width M1 = width M2 ; :: thesis: verum

end;

end;

end;

hence Indices M1 = Indices M2 by A2; :: thesis: verum