reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th36:
  for p being Point of TOP-REAL 2 st p = CircleMap.r holds
  Arg(p) = 2*PI*frac(r)
  proof
    let p be Point of T2;
    set z = euc2cpx(p);
    set A = 2*PI*frac(r);
    assume
A1: p = CM.r;
    reconsider q = CM.R^1(r) as Point of T2 by PRE_TOPC:25;
A2: |.z.| = |.p.| by EUCLID_3:25;
A3: |.q.| = 1 by TOPREALB:12;
    frac r < 1 & 0 <= frac r by INT_1:43;
    then
A4: 2*PI*Q <= A & A < 2*PI*1 by XREAL_1:68;
A6: CM.r = |[cos(2*PI*r),sin(2*PI*r)]| by TOPREALB:def 11;
    A = 2*PI*r + 2*PI*(-[\r/]);
    then cos(2*PI*r) = cos A & sin(2*PI*r) = sin A by COMPLEX2:8,9;
    then z = |.z.|*cos A + |.z.|*sin A*<i> by A1,A2,A3,A6;
    hence thesis by A4,A2,A1,A3,COMPLEX1:44,COMPTRIG:def 1;
  end;
