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 )

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

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

end;( 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 A5, TARSKI:def 2;

end;

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 )

( 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 A2, FINSEQ_1:44;

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 A2, A4, A6, FINSEQ_1:44; :: thesis: verum

end;reconsider z = p . 1 as Element of H by A2, FINSEQ_1:44;

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 A2, A4, A6, FINSEQ_1:44; :: thesis: verum

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 )

( 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 A3, FINSEQ_1:44;

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 A3, A1, A7, FINSEQ_1:44; :: thesis: verum

end;reconsider z = p . 2 as Element of H by A3, FINSEQ_1:44;

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 A3, A1, A7, FINSEQ_1:44; :: thesis: verum