reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem
  frac r = 3/4 implies CircleMap.r = |[0,-1]|
proof
  assume frac r = 3/4;
  then
A1: r-[\r/] = 3/4 by INT_1:def 8;
  thus CircleMap.r = CircleMap.(r+-[\r/]) by Th31
    .= |[ cos(2*PI*(3/4)), sin(2*PI*(3/4)) ]| by A1,Def11
    .= |[0,-1]| by SIN_COS:77;
end;
