let G1, G2 be Group; :: thesis: ( G1 is Subgroup of G2 & G2 is Subgroup of G1 implies multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) )

assume that

A1: G1 is Subgroup of G2 and

A2: G2 is Subgroup of G1 ; :: thesis: multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #)

set g = the multF of G2;

set f = the multF of G1;

set B = the carrier of G2;

set A = the carrier of G1;

A3: ( the carrier of G1 c= the carrier of G2 & the carrier of G2 c= the carrier of G1 ) by A1, A2, Def5;

then A4: the carrier of G1 = the carrier of G2 ;

the multF of G1 = the multF of G2 || the carrier of G1 by A1, Def5

.= ( the multF of G1 || the carrier of G2) || the carrier of G1 by A2, Def5

.= the multF of G1 || the carrier of G2 by A4, RELAT_1:72

.= the multF of G2 by A2, Def5 ;

hence multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) by A3, XBOOLE_0:def 10; :: thesis: verum

assume that

A1: G1 is Subgroup of G2 and

A2: G2 is Subgroup of G1 ; :: thesis: multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #)

set g = the multF of G2;

set f = the multF of G1;

set B = the carrier of G2;

set A = the carrier of G1;

A3: ( the carrier of G1 c= the carrier of G2 & the carrier of G2 c= the carrier of G1 ) by A1, A2, Def5;

then A4: the carrier of G1 = the carrier of G2 ;

the multF of G1 = the multF of G2 || the carrier of G1 by A1, Def5

.= ( the multF of G1 || the carrier of G2) || the carrier of G1 by A2, Def5

.= the multF of G1 || the carrier of G2 by A4, RELAT_1:72

.= the multF of G2 by A2, Def5 ;

hence multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) by A3, XBOOLE_0:def 10; :: thesis: verum