let G be non empty multMagma ; :: thesis: ( G is right-cancelable iff for a, b, c being Element of G st b * a = c * a holds
b = c )
thus ( G is right-cancelable implies for a, b, c being Element of G st b * a = c * a holds
b = c ) :: thesis: ( ( for a, b, c being Element of G st b * a = c * a holds
b = c ) implies G is right-cancelable )

proof

assume A1: for a, b, c being Element of G st H₂(G) . (b,a) = H₂(G) . (c,a) holds
b = c ; :: according to MONOID_0:def 7,MONOID_0:def 18 :: thesis: for a, b, c being Element of G st b * a = c * a holds
b = c
let a, b, c be Element of G; :: thesis: ( b * a = c * a implies b = c )
thus ( b * a = c * a implies b = c ) by A1; :: thesis: verum

end;

assume A2: for a, b, c being Element of G st b * a = c * a holds
b = c ; :: thesis: G is right-cancelable
let a be Element of G; :: according to MONOID_0:def 7,MONOID_0:def 18 :: thesis: for b, c being Element of the carrier of G st the multF of G . (b,a) = the multF of G . (c,a) holds
b = c
let b, c be Element of G; :: thesis: ( the multF of G . (b,a) = the multF of G . (c,a) implies b = c )
( b * a = H₂(G) . (b,a) & c * a = H₂(G) . (c,a) ) ;
hence ( the multF of G . (b,a) = the multF of G . (c,a) implies b = c ) by A2; :: thesis: verum