reserve E,F,G for RealNormSpace;
reserve f for Function of E,F;
reserve g for Function of F,G;
reserve a,b,c for Point of E;
reserve t for Real;

theorem Th18: :: a is the only fixed point of R
  (a-reflection).a = a & for b st (a-reflection).b = b holds a = b
  proof
    set R = a-reflection;
    thus R.a = 2*a - a by Def4
    .= a + a - a by Th3
    .= a + (a - a) by RLVECT_1:28
    .= a + 0.E by RLVECT_1:15
    .= a;
    let b;
    assume R.b = b;
    then
A1: R.b + b = 2*b by Th3;
    R.b = 2*a - b by Def4;
    then R.b + b = 2*a - (b - b) by RLVECT_1:29
    .= 2*a - 0.E by RLVECT_1:15
    .= 2*a;
    hence thesis by A1,RLVECT_1:36;
  end;
