
theorem Th54:
  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, O being Point of I[01] st a<b
& h=AffineMap(2/(b-a),-(b+a)/(b-a),2/(d-c),-(d+c)/(d-c)) & (f.O)`1=a holds ((h*
  f).O)`1= -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, O 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: a<b and
A2: h=AffineMap(A,B,C,D) and
A3: (f.O)`1=a;
A4: b-a>0 by A1,XREAL_1:50;
  dom f=the carrier of I[01] by FUNCT_2:def 1;
  then
A5: ((h*f).O)=(h.(f.O)) by FUNCT_1:13;
A6: (h.(f.O))= |[A*((f.O)`1)+B,C*((f.O)`2)+D]| by A2,JGRAPH_2:def 2;
  A*((f.O)`1)+B = (2*a)/(b-a)+ -(b+a)/(b-a) by A3,XCMPLX_1:74
    .= (2*a)/(b-a)+ (-(b+a))/(b-a) by XCMPLX_1:187
    .=(2*a+-(b+a))/(b-a) by XCMPLX_1:62
    .=(-(b-a))/(b-a)
    .= -1 by A4,XCMPLX_1:197;
  hence thesis by A5,A6,EUCLID:52;
end;
