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).:[.0,1.] = [.b,a+b.]
proof
  let a,b be Real such that
A1: a > 0;
  thus AffineMap(a,b).:[.0,1.] c= [.b,a+b.]
  proof
A2: AffineMap(a,b).1 = a+b by Th49;
    let u be object;
    assume u in AffineMap(a,b).:[.0,1.];
    then consider r being Element of REAL such that
A3: r in [.0,1.] and
A4: u = AffineMap(a,b).r by FUNCT_2:65;
    reconsider s = u as Real by A4;
    r <= 1 by A3,XXREAL_1:1;
    then
A5: s <= a+b by A1,A4,A2,Th53;
A6: AffineMap(a,b).0 = b by Th48;
    0 <= r by A3,XXREAL_1:1;
    then b <= s by A1,A4,A6,Th53;
    hence thesis by A5,XXREAL_1:1;
  end;
  let u be object;
  assume
A7: u in [.b,a+b.];
  then reconsider r = u as Element of REAL;
  set s = (r - b)/a;
A8: AffineMap(a,b).s = a*s + b by Def4
    .= r - b + b by A1,XCMPLX_1:87
    .= r;
  r <= a+b by A7,XXREAL_1:1;
  then r-b <= a by XREAL_1:20;
  then s <= a/a by A1,XREAL_1:72;
  then
A9: s <= 1 by A1,XCMPLX_1:60;
  b <= r by A7,XXREAL_1:1;
  then 0 <= r - b by XREAL_1:48;
  then s in [.0,1.] by A1,A9,XXREAL_1:1;
  hence thesis by A8,FUNCT_2:35;
end;
