F1 ^ F2 is Matrix-yielding

proof

let i be Nat; :: according to MATRIXJ1:def 2 :: thesis: ( i in dom (F1 ^ F2) implies (F1 ^ F2) . i is Matrix of D )
assume A1: i in dom (F1 ^ F2) ; :: thesis: (F1 ^ F2) . i is Matrix of D

per cases ( i in dom F1 or ex n being Nat st
( n in dom F2 & i = (len F1) + n ) ) by A1, FINSEQ_1:25;

suppose i in dom F1 ; :: thesis: (F1 ^ F2) . i is Matrix of D

then (F1 ^ F2) . i = F1 . i by FINSEQ_1:def 7;
hence (F1 ^ F2) . i is Matrix of D ; :: thesis: verum

end;

suppose ex n being Nat st
( n in dom F2 & i = (len F1) + n ) ; :: thesis: (F1 ^ F2) . i is Matrix of D

then consider n being Nat such that
A2: n in dom F2 and
A3: i = (len F1) + n ;
(F1 ^ F2) . i = F2 . n by A2, A3, FINSEQ_1:def 7;
hence (F1 ^ F2) . i is Matrix of D ; :: thesis: verum

end;

end;

end;

hence F1 ^ F2 is FinSequence_of_Matrix of D ; :: thesis: verum