let G be Group; :: thesis: for H1, H2 being Subgroup of G st the carrier of H1 = the carrier of H2 holds

multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #)

let H1, H2 be Subgroup of G; :: thesis: ( the carrier of H1 = the carrier of H2 implies multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #) )

assume the carrier of H1 = the carrier of H2 ; :: thesis: multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #)

then ( H1 is Subgroup of H2 & H2 is Subgroup of H1 ) by Th57;

hence multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #) by Th55; :: thesis: verum

multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #)

let H1, H2 be Subgroup of G; :: thesis: ( the carrier of H1 = the carrier of H2 implies multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #) )

assume the carrier of H1 = the carrier of H2 ; :: thesis: multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #)

then ( H1 is Subgroup of H2 & H2 is Subgroup of H1 ) by Th57;

hence multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #) by Th55; :: thesis: verum