let G1, G2 be Group; :: thesis: for x being Element of G1
for y being Element of G2 holds <*x,y*> " = <*(x "),(y ")*>

let x be Element of G1; :: thesis: for y being Element of G2 holds <*x,y*> " = <*(x "),(y ")*>
let y be Element of G2; :: thesis: <*x,y*> " = <*(x "),(y ")*>
set G = <*G1,G2*>;
A1: <*G1,G2*> . 2 = G2 by FINSEQ_1:44;
reconsider lF = <*x,y*>, p = <*(x "),(y ")*> as Element of product (Carrier <*G1,G2*>) by Def2;
A2: p . 1 = x " by FINSEQ_1:44;
A3: p . 2 = y " by FINSEQ_1:44;
A4: <*G1,G2*> . 1 = G1 by FINSEQ_1:44;
for i being set st i in {1,2} holds
ex H being Group ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )
proof
let i be set ; :: thesis: ( i in {1,2} implies ex H being Group ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i ) )

assume A5: i in {1,2} ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )

per cases ( i = 1 or i = 2 ) by ;
suppose A6: i = 1 ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )

reconsider H = <*G1,G2*> . 1 as Group by FINSEQ_1:44;
reconsider z = p . 1 as Element of H by ;
take H ; :: thesis: ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )

take z " ; :: thesis: ( H = <*G1,G2*> . i & p . i = (z ") " & z " = lF . i )
thus H = <*G1,G2*> . i by A6; :: thesis: ( p . i = (z ") " & z " = lF . i )
thus p . i = (z ") " by A6; :: thesis: z " = lF . i
thus z " = lF . i by ; :: thesis: verum
end;
suppose A7: i = 2 ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )

reconsider H = <*G1,G2*> . 2 as Group by FINSEQ_1:44;
reconsider z = p . 2 as Element of H by ;
take H ; :: thesis: ex z being Element of H st
( H = <*G1,G2*> . i & p . i = z " & z = lF . i )

take z " ; :: thesis: ( H = <*G1,G2*> . i & p . i = (z ") " & z " = lF . i )
thus H = <*G1,G2*> . i by A7; :: thesis: ( p . i = (z ") " & z " = lF . i )
thus p . i = (z ") " by A7; :: thesis: z " = lF . i
thus z " = lF . i by ; :: thesis: verum
end;
end;
end;
hence <*x,y*> " = <*(x "),(y ")*> by Th7; :: thesis: verum