let G1, G2 be non empty Group-like multMagma ; :: thesis: 1_ (product <*G1,G2*>) = <*(1_ G1),(1_ G2)*>
set s = <*(1_ G1),(1_ G2)*>;
set f = <*G1,G2*>;
dom <*(1_ G1),(1_ G2)*> = {1,2} by ;
then reconsider s = <*(1_ G1),(1_ G2)*> as ManySortedSet of {1,2} by ;
for i being set st i in {1,2} holds
ex G being non empty Group-like multMagma st
( G = <*G1,G2*> . i & s . i = 1_ G )
proof
let i be set ; :: thesis: ( i in {1,2} implies ex G being non empty Group-like multMagma st
( G = <*G1,G2*> . i & s . i = 1_ G ) )

assume A1: i in {1,2} ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2*> . i & s . i = 1_ G )

per cases ( i = 1 or i = 2 ) by ;
suppose A2: i = 1 ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2*> . i & s . i = 1_ G )

then reconsider G = <*G1,G2*> . i as non empty Group-like multMagma by FINSEQ_1:44;
take G ; :: thesis: ( G = <*G1,G2*> . i & s . i = 1_ G )
<*G1,G2*> . i = G1 by ;
hence ( G = <*G1,G2*> . i & s . i = 1_ G ) by ; :: thesis: verum
end;
suppose A3: i = 2 ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2*> . i & s . i = 1_ G )

then reconsider G = <*G1,G2*> . i as non empty Group-like multMagma by FINSEQ_1:44;
take G ; :: thesis: ( G = <*G1,G2*> . i & s . i = 1_ G )
<*G1,G2*> . i = G2 by ;
hence ( G = <*G1,G2*> . i & s . i = 1_ G ) by ; :: thesis: verum
end;
end;
end;
hence 1_ (product <*G1,G2*>) = <*(1_ G1),(1_ G2)*> by Th5; :: thesis: verum