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

proof

assume A1: for a, b, c being Element of G st ( H₂(G) . a,b = H₂(G) . a,c or H₂(G) . b,a = H₂(G) . c,a ) holds
b = c ; :: according to MONOID_0:def 8,MONOID_0:def 19 :: thesis: for a, b, c being Element of G st ( a * b = a * c or b * a = c * a ) holds
b = c
let a, b, c be Element of G; :: thesis: ( ( a * b = a * c or b * a = c * a ) implies b = c )
thus ( ( a * b = a * c or b * a = c * a ) implies b = c ) by A1; :: thesis: verum

end;

assume A2: for a, b, c being Element of G st ( a * b = a * c or b * a = c * a ) holds
b = c ; :: thesis: G is cancelable
let a be Element of G; :: according to MONOID_0:def 8,MONOID_0:def 19 :: thesis: for b, c being Element of the carrier of G st ( the multF of G . a,b = the multF of G . a,c or 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 . a,b = the multF of G . a,c or the multF of G . b,a = the multF of G . c,a ) implies b = c )
( a * b = H₂(G) . a,b & a * c = H₂(G) . a,c & b * a = H₂(G) . b,a & c * a = H₂(G) . c,a ) ;
hence ( ( the multF of G . a,b = the multF of G . a,c or the multF of G . b,a = the multF of G . c,a ) implies b = c ) by A2; :: thesis: verum