let G1 be non empty multMagma ; :: thesis: <*G1*> is multMagma-Family of {1}

dom <*G1*> = {1} by FINSEQ_1:2, FINSEQ_1:def 8;

then reconsider A = <*G1*> as ManySortedSet of {1} by PARTFUN1:def 2, RELAT_1:def 18;

A is multMagma-yielding

dom <*G1*> = {1} by FINSEQ_1:2, FINSEQ_1:def 8;

then reconsider A = <*G1*> as ManySortedSet of {1} by PARTFUN1:def 2, RELAT_1:def 18;

A is multMagma-yielding

proof

hence
<*G1*> is multMagma-Family of {1}
; :: thesis: verum
let y be set ; :: according to GROUP_7:def 1 :: thesis: ( y in rng A implies y is non empty multMagma )

assume y in rng A ; :: thesis: y is non empty multMagma

then consider x being object such that

A1: x in dom A and

A2: A . x = y by FUNCT_1:def 3;

x = 1 by A1, TARSKI:def 1;

hence y is non empty multMagma by A2, FINSEQ_1:def 8; :: thesis: verum

end;assume y in rng A ; :: thesis: y is non empty multMagma

then consider x being object such that

A1: x in dom A and

A2: A . x = y by FUNCT_1:def 3;

x = 1 by A1, TARSKI:def 1;

hence y is non empty multMagma by A2, FINSEQ_1:def 8; :: thesis: verum