let G be addGroup; :: thesis: for H1, H2 being Subgroup of G st ( for g being Element of G holds
( g in H1 iff g in H2 ) ) holds
addMagma(# the carrier of H1, the addF of H1 #) = addMagma(# the carrier of H2, the addF of H2 #)
let H1, H2 be Subgroup of G; :: thesis: ( ( for g being Element of G holds
( g in H1 iff g in H2 ) ) implies addMagma(# the carrier of H1, the addF of H1 #) = addMagma(# the carrier of H2, the addF of H2 #) )
assume for g being Element of G holds
( g in H1 iff g in H2 ) ; :: thesis: addMagma(# the carrier of H1, the addF of H1 #) = addMagma(# the carrier of H2, the addF of H2 #)
then ( H1 is Subgroup of H2 & H2 is Subgroup of H1 ) by Th58;
hence addMagma(# the carrier of H1, the addF of H1 #) = addMagma(# the carrier of H2, the addF of H2 #) by Th55; :: thesis: verum