 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;

theorem
  G is right-cancelable iff for a,b,c st b*a = c*a holds b = c
proof
  thus G is right-cancelable implies for a,b,c st b*a = c*a holds b = c
  proof
    assume
A1: for a,b,c st op(G).(b,a) = op(G).(c,a) holds b = c;
    let a,b,c;
    thus thesis by A1;
  end;
  assume
A2: for a,b,c st b*a = c*a holds b = c;
  let a,b,c;
  b*a = op(G).(b,a) & c*a = op(G).(c,a);
  hence thesis by A2;
end;
