 reserve a,b,c,x for Real;

theorem Hope4:
  c - b <> 0 & AffineMap (-1/(c-b),c/(c-b)).x = 1 implies
    x = b
  proof
    assume that
A0: c - b <> 0 and
A1: AffineMap (-1/(c-b),c/(c-b)).x = 1;
    x in REAL & b in REAL by XREAL_0:def 1; then
A3: x in dom AffineMap (-1/(c-b),c/(c-b)) &
      b in dom AffineMap (-1/(c-b),c/(c-b)) by FUNCT_2:def 1;
    AffineMap (-1/(c-b),c/(c-b)).b = 1 by A0,Cb1;
    hence thesis by A1,FUNCT_1:def 4,A3,A0;
  end;
