set xy = <*x,y,z*>;
set G = <*G1,G2,G3*>;
A1: dom (Carrier <*G1,G2,G3*>) = {1,2,3} by PARTFUN1:def 2;
A8: for a being object st a in {1,2,3} holds
<*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a

proof

let a be object ; :: thesis: ( a in {1,2,3} implies <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a )
assume A9: a in {1,2,3} ; :: thesis: <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a

per cases ( a = 1 or a = 2 or a = 3 ) by A9, ENUMSET1:def 1;

suppose A10: a = 1 ; :: thesis: <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a

then ex R being 1-sorted st
( R = <*G1,G2,G3*> . 1 & (Carrier <*G1,G2,G3*>) . 1 = the carrier of R ) by A9, PRALG_1:def 15;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A10; :: thesis: verum

end;

suppose A11: a = 2 ; :: thesis: <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a

then ex R being 1-sorted st
( R = <*G1,G2,G3*> . 2 & (Carrier <*G1,G2,G3*>) . 2 = the carrier of R ) by A9, PRALG_1:def 15;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A11; :: thesis: verum

end;

suppose A12: a = 3 ; :: thesis: <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a

then ex R being 1-sorted st
( R = <*G1,G2,G3*> . 3 & (Carrier <*G1,G2,G3*>) . 3 = the carrier of R ) by A9, PRALG_1:def 15;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A12; :: thesis: verum

end;

end;

end;

dom <*x,y,z*> = {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then <*x,y,z*> in product (Carrier <*G1,G2,G3*>) by A1, A8, CARD_3:9;
hence <*x,y,z*> is Element of (product <*G1,G2,G3*>) by Def2; :: thesis: verum