 reserve a,b,c,x for Real;

theorem Hope3:
  b - a <> 0 & AffineMap (1/(b-a),-a/(b-a)).x = 1 implies
    x = b
  proof
    assume that
A0: b - a <> 0 and
A1: AffineMap (1/(b-a),-a/(b-a)).x = 1;
    x in REAL & b in REAL by XREAL_0:def 1; then
A3: x in dom AffineMap (1/(b-a),-a/(b-a)) &
      b in dom AffineMap (1/(b-a),-a/(b-a)) by FUNCT_2:def 1;
    AffineMap (1/(b-a),-a/(b-a)).b = 1 by A0,Ab1;
    hence thesis by A1,FUNCT_1:def 4,A3,A0;
  end;
