S1 ^ S2 is Square-Matrix-yielding

proof

let i be Nat; :: according to MATRIXJ1:def 6 :: thesis: ( i in dom (S1 ^ S2) implies ex n being Nat st (S1 ^ S2) . i is Matrix of n,D )
assume A1: i in dom (S1 ^ S2) ; :: thesis: ex n being Nat st (S1 ^ S2) . i is Matrix of n,D
take len ((S1 ^ S2) . i) ; :: thesis: (S1 ^ S2) . i is Matrix of len ((S1 ^ S2) . i),D

per cases ( i in dom S1 or ex n being Nat st
( n in dom S2 & i = (len S1) + n ) ) by A1, FINSEQ_1:25;

suppose i in dom S1 ; :: thesis: (S1 ^ S2) . i is Matrix of len ((S1 ^ S2) . i),D

then (S1 ^ S2) . i = S1 . i by FINSEQ_1:def 7;
hence (S1 ^ S2) . i is Matrix of len ((S1 ^ S2) . i),D ; :: thesis: verum

end;

suppose ex n being Nat st
( n in dom S2 & i = (len S1) + n ) ; :: thesis: (S1 ^ S2) . i is Matrix of len ((S1 ^ S2) . i),D

then consider n being Nat such that
A2: n in dom S2 and
A3: i = (len S1) + n ;
(S1 ^ S2) . i = S2 . n by A2, A3, FINSEQ_1:def 7;
hence (S1 ^ S2) . i is Matrix of len ((S1 ^ S2) . i),D ; :: thesis: verum

end;

end;

end;

hence S1 ^ S2 is FinSequence_of_Square-Matrix of D ; :: thesis: verum