
theorem Th55:
  for a,b,c,d being Real, h being Function of TOP-REAL 2,
TOP-REAL 2,f being Function of I[01],TOP-REAL 2, I being Point of I[01] st c <d
& h=AffineMap(2/(b-a),-(b+a)/(b-a),2/(d-c),-(d+c)/(d-c)) & (f.I)`2=d holds ((h*
  f).I)`2= 1
proof
  let a,b,c,d be Real, h be Function of TOP-REAL 2,TOP-REAL 2,f be
  Function of I[01],TOP-REAL 2, I be Point of I[01];
  set A=2/(b-a), B=-(b+a)/(b-a), C = 2/(d-c), D=-(d+c)/(d-c);
  assume that
A1: c <d and
A2: h=AffineMap(A,B,C,D) and
A3: (f.I)`2=d;
A4: d-c >0 by A1,XREAL_1:50;
  dom f=the carrier of I[01] by FUNCT_2:def 1;
  then
A5: ((h*f).I)=(h.(f.I)) by FUNCT_1:13;
A6: (h.(f.I))= |[A*((f.I)`1)+B,C*((f.I)`2)+D]| by A2,JGRAPH_2:def 2;
  C*((f.I)`2)+D = (2*d)/(d-c)+ -(d+c)/(d-c) by A3,XCMPLX_1:74
    .= (2*d)/(d-c)+ (-(d+c))/(d-c) by XCMPLX_1:187
    .=(2*d+-(d+c))/(d-c) by XCMPLX_1:62
    .= 1 by A4,XCMPLX_1:60;
  hence thesis by A5,A6,EUCLID:52;
end;
