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 Th41:
  for s being Real, p being Point of TOP-REAL 2 holds
  |.(Rotate(s)).p.| = |.p.|
  proof
    let s be Real;
    let p be Point of T2;
    set c = euc2cpx(p);
    set q = (Rotate(s)).p;
A1: Re Rotate(c,s) = |.c.|*cos (s+Arg c) &
    Im Rotate(c,s) = |.c.|*sin (s+Arg c) by COMPLEX1:12;
    q = cpx2euc(Rotate(c,s)) by JORDAN24:def 3;
    then
A2: q`1 = Re Rotate(c,s) & q`2 = Im Rotate(c,s);
    |.p.|^2 = |.c.|^2*1 by EUCLID_3:25
    .= |.c.|^2*((cos(s+Arg c))^2 + (sin(s+Arg c))^2) by SIN_COS:29
    .= (|.c.|*cos(s+Arg c))^2 + (|.c.|*sin(s+Arg c))^2
    .= |.q.|^2 by A1,A2,JGRAPH_1:29;
    hence |.p.| = |.q.| by Th1;
  end;
