let G1, G2 be addGroup; :: thesis: ( G1 is Subgroup of G2 & G2 is Subgroup of G1 implies addMagma(# the carrier of G1, the addF of G1 #) = addMagma(# the carrier of G2, the addF of G2 #) )
assume that
A1: G1 is Subgroup of G2 and
A2: G2 is Subgroup of G1 ; :: thesis: addMagma(# the carrier of G1, the addF of G1 #) = addMagma(# the carrier of G2, the addF of G2 #)
set g = the addF of G2;
set f = the addF 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, DefA5;
A4: the carrier of G1 = the carrier of G2 by A1, A2, DefA5;
the addF of G1 = the addF of G2 || the carrier of G1 by A1, DefA5
.= ( the addF of G1 || the carrier of G2) || the carrier of G1 by A2, DefA5
.= the addF of G1 || the carrier of G2 by A4, RELAT_1:72
.= the addF of G2 by A2, DefA5 ;
hence addMagma(# the carrier of G1, the addF of G1 #) = addMagma(# the carrier of G2, the addF of G2 #) by A3, XBOOLE_0:def 10; :: thesis: verum