reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  for a,b being Real st a <> 0 holds (AffineMap(a,b) qua
  Function)" = AffineMap(a",-b/a)
proof
  let a,b be Real such that
A1: a <> 0;
  for x being Element of REAL holds (AffineMap(a",-b/a)*AffineMap(a,b)).x
  = (id REAL).x
  proof
    let x being Element of REAL;
    thus (AffineMap(a",-b/a)*AffineMap(a,b)).x = AffineMap(a",-b/a).(AffineMap
    (a,b).x) by FUNCT_2:15
      .= AffineMap(a",-b/a).(a*x+b) by Def4
      .= a"*(a*x+b)+-b/a by Def4
      .= a"*a*x+a"*b +-b/a
      .= 1 *x+a"*b +-b/a by A1,XCMPLX_0:def 7
      .= x+a"*b -b/a
      .= x+ b/a -b/a by XCMPLX_0:def 9
      .= (id REAL).x;
  end;
  then
A2: AffineMap(a",-b/a)*AffineMap(a,b) = id REAL by FUNCT_2:63;
  rng AffineMap(a,b) = REAL by A1,Th55;
  hence thesis by A1,A2,Th50,FUNCT_2:30;
end;
