let G1, G2, G3 be Group; :: thesis: for x being Element of G1
for y being Element of G2
for z being Element of G3 holds <*x,y,z*> " = <*(x "),(y "),(z ")*>
let x be Element of G1; :: thesis: for y being Element of G2
for z being Element of G3 holds <*x,y,z*> " = <*(x "),(y "),(z ")*>
let y be Element of G2; :: thesis: for z being Element of G3 holds <*x,y,z*> " = <*(x "),(y "),(z ")*>
let z be Element of G3; :: thesis: <*x,y,z*> " = <*(x "),(y "),(z ")*>
set G = <*G1,G2,G3*>;
reconsider lF = <*x,y,z*>, p = <*(x "),(y "),(z ")*> as Element of product (Carrier <*G1,G2,G3*>) by Def2;
for i being set st i in {1,2,3} holds
ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )

proof

let i be set ; :: thesis: ( i in {1,2,3} implies ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i ) )
assume A7: i in {1,2,3} ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )

per cases ( i = 1 or i = 2 or i = 3 ) by A7, ENUMSET1:def 1;

suppose A8: i = 1 ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )

reconsider H = <*G1,G2,G3*> . 1 as Group ;
reconsider z = p . 1 as Element of H ;
take H ; :: thesis: ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )
take z " ; :: thesis: ( H = <*G1,G2,G3*> . i & p . i = (z ") " & z " = lF . i )
thus H = <*G1,G2,G3*> . i by A8; :: thesis: ( p . i = (z ") " & z " = lF . i )
thus p . i = (z ") " by A8; :: thesis: z " = lF . i
thus z " = lF . i by A8; :: thesis: verum

end;

suppose A9: i = 2 ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )

reconsider H = <*G1,G2,G3*> . 2 as Group ;
reconsider z = p . 2 as Element of H ;
take H ; :: thesis: ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )
take z " ; :: thesis: ( H = <*G1,G2,G3*> . i & p . i = (z ") " & z " = lF . i )
thus H = <*G1,G2,G3*> . i by A9; :: thesis: ( p . i = (z ") " & z " = lF . i )
thus p . i = (z ") " by A9; :: thesis: z " = lF . i
thus z " = lF . i by A9; :: thesis: verum

end;

suppose A10: i = 3 ; :: thesis: ex H being Group ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )

reconsider H = <*G1,G2,G3*> . 3 as Group ;
reconsider z = p . 3 as Element of H ;
take H ; :: thesis: ex z being Element of H st
( H = <*G1,G2,G3*> . i & p . i = z " & z = lF . i )
take z " ; :: thesis: ( H = <*G1,G2,G3*> . i & p . i = (z ") " & z " = lF . i )
thus H = <*G1,G2,G3*> . i by A10; :: thesis: ( p . i = (z ") " & z " = lF . i )
thus p . i = (z ") " by A10; :: thesis: z " = lF . i
thus z " = lF . i by A10; :: thesis: verum

end;

end;

end;

hence <*x,y,z*> " = <*(x "),(y "),(z ")*> by Th7; :: thesis: verum