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 Th55:
  for a,b being Real st a <> 0 holds rng AffineMap(a,b) = REAL
proof
  let a,b be Real such that
A1: a <> 0;
  thus rng AffineMap(a,b) c= REAL;
  let e be object;
  assume e in REAL;
  then reconsider r = e as Element of REAL;
  reconsider s = (r - b)/a as Element of REAL by XREAL_0:def 1;
  AffineMap(a,b).s = a*s + b by Def4
    .= r - b + b by A1,XCMPLX_1:87
    .= r;
   then r in rng AffineMap(a,b) by FUNCT_2:4;
  hence thesis;
end;
