let G1, G2, G3 be non empty Group-like multMagma ; :: thesis: 1_ (product <*G1,G2,G3*>) = <*(1_ G1),(1_ G2),(1_ G3)*>
set s = <*(1_ G1),(1_ G2),(1_ G3)*>;
set f = <*G1,G2,G3*>;
<*(1_ G1),(1_ G2),(1_ G3)*> is ManySortedSet of

proof

dom <*(1_ G1),(1_ G2),(1_ G3)*> = {1,2,3} by FINSEQ_3:1, FINSEQ_3:30;
hence <*(1_ G1),(1_ G2),(1_ G3)*> is ManySortedSet of by PARTFUN1:def 4, RELAT_1:def 18; :: thesis: verum

end;

then reconsider s = <*(1_ G1),(1_ G2),(1_ G3)*> as ManySortedSet of ;
for i being set st i in {1,2,3} holds
ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G )

proof

let i be set ; :: thesis: ( i in {1,2,3} implies ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G ) )
assume A1: i in {1,2,3} ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G )

per cases ( i = 1 or i = 2 or i = 3 ) by A1, ENUMSET1:def 1;

suppose A2: i = 1 ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G )

then reconsider G = <*G1,G2,G3*> . i as non empty Group-like multMagma by FINSEQ_1:62;
take G ; :: thesis: ( G = <*G1,G2,G3*> . i & s . i = 1_ G )
<*G1,G2,G3*> . i = G1 by A2, FINSEQ_1:62;
hence ( G = <*G1,G2,G3*> . i & s . i = 1_ G ) by A2, FINSEQ_1:62; :: thesis: verum

end;

suppose A3: i = 2 ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G )

then reconsider G = <*G1,G2,G3*> . i as non empty Group-like multMagma by FINSEQ_1:62;
take G ; :: thesis: ( G = <*G1,G2,G3*> . i & s . i = 1_ G )
<*G1,G2,G3*> . i = G2 by A3, FINSEQ_1:62;
hence ( G = <*G1,G2,G3*> . i & s . i = 1_ G ) by A3, FINSEQ_1:62; :: thesis: verum

end;

suppose A4: i = 3 ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1,G2,G3*> . i & s . i = 1_ G )

then reconsider G = <*G1,G2,G3*> . i as non empty Group-like multMagma by FINSEQ_1:62;
take G ; :: thesis: ( G = <*G1,G2,G3*> . i & s . i = 1_ G )
<*G1,G2,G3*> . i = G3 by A4, FINSEQ_1:62;
hence ( G = <*G1,G2,G3*> . i & s . i = 1_ G ) by A4, FINSEQ_1:62; :: thesis: verum

end;

end;

end;

hence 1_ (product <*G1,G2,G3*>) = <*(1_ G1),(1_ G2),(1_ G3)*> by Th8; :: thesis: verum