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

proof

dom <*(1_ G1)*> = {1} by FINSEQ_1:4, FINSEQ_1:def 8;
hence <*(1_ G1)*> is ManySortedSet of by PARTFUN1:def 4, RELAT_1:def 18; :: thesis: verum

end;

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

proof

let i be set ; :: thesis: ( i in {1} implies ex G being non empty Group-like multMagma st
( G = <*G1*> . i & s . i = 1_ G ) )
assume i in {1} ; :: thesis: ex G being non empty Group-like multMagma st
( G = <*G1*> . i & s . i = 1_ G )
then A1: i = 1 by TARSKI:def 1;
then reconsider G = <*G1*> . i as non empty Group-like multMagma by FINSEQ_1:def 8;
take G ; :: thesis: ( G = <*G1*> . i & s . i = 1_ G )
<*G1*> . i = G1 by A1, FINSEQ_1:def 8;
hence ( G = <*G1*> . i & s . i = 1_ G ) by A1, FINSEQ_1:def 8; :: thesis: verum

end;

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