set xy = <*x,y,z*>;
set G = <*G1,G2,G3*>;
A1: dom (Carrier <*G1,G2,G3*>) = {1,2,3} by PARTFUN1:def 2;
A2: <*x,y,z*> . 2 = y by FINSEQ_1:45;
A3: <*G1,G2,G3*> . 1 = G1 by FINSEQ_1:45;
A4: <*x,y,z*> . 3 = z by FINSEQ_1:45;
A5: <*G1,G2,G3*> . 3 = G3 by FINSEQ_1:45;
A6: <*G1,G2,G3*> . 2 = G2 by FINSEQ_1:45;
A7: <*x,y,z*> . 1 = x by FINSEQ_1:45;
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 ;
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 ;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A7, A3, 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 ;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A2, A6, 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 ;
hence <*x,y,z*> . a in (Carrier <*G1,G2,G3*>) . a by A4, A5, A12; :: thesis: verum
end;
end;
end;
dom <*x,y,z*> = {1,2,3} by ;
then <*x,y,z*> in product (Carrier <*G1,G2,G3*>) by ;
hence <*x,y,z*> is Element of (product <*G1,G2,G3*>) by Def2; :: thesis: verum