set xy = <*x,y*>;
set G = <*G1,G2*>;
A1: dom (Carrier <*G1,G2*>) = {1,2} by PARTFUN1:def 2;
A2: <*x,y*> . 2 = y by FINSEQ_1:44;
A3: <*G1,G2*> . 2 = G2 by FINSEQ_1:44;
A4: <*G1,G2*> . 1 = G1 by FINSEQ_1:44;
A5: <*x,y*> . 1 = x by FINSEQ_1:44;
A6: for a being object st a in {1,2} holds
<*x,y*> . a in (Carrier <*G1,G2*>) . a
proof
let a be object ; :: thesis: ( a in {1,2} implies <*x,y*> . a in (Carrier <*G1,G2*>) . a )
assume A7: a in {1,2} ; :: thesis: <*x,y*> . a in (Carrier <*G1,G2*>) . a
per cases ( a = 1 or a = 2 ) by ;
suppose A8: a = 1 ; :: thesis: <*x,y*> . a in (Carrier <*G1,G2*>) . a
then ex R being 1-sorted st
( R = <*G1,G2*> . 1 & (Carrier <*G1,G2*>) . 1 = the carrier of R ) by ;
hence <*x,y*> . a in (Carrier <*G1,G2*>) . a by A5, A4, A8; :: thesis: verum
end;
suppose A9: a = 2 ; :: thesis: <*x,y*> . a in (Carrier <*G1,G2*>) . a
then ex R being 1-sorted st
( R = <*G1,G2*> . 2 & (Carrier <*G1,G2*>) . 2 = the carrier of R ) by ;
hence <*x,y*> . a in (Carrier <*G1,G2*>) . a by A2, A3, A9; :: thesis: verum
end;
end;
end;
dom <*x,y*> = {1,2} by ;
then <*x,y*> in product (Carrier <*G1,G2*>) by ;
hence <*x,y*> is Element of (product <*G1,G2*>) by Def2; :: thesis: verum