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 Th52:
  eLoop(r) = SphereMap * ExtendInt(r)
  proof
    let x be Point of I[01];
A1: (ExtendInt(r)).x = r*x by TOPALG_5:def 1;
    thus (eLoop(r)).x = |[ cos(2*PI*r*x), sin(2*PI*r*x), 0 ]| by Def7
    .= |[ cos(2*PI*(ExtendInt(r)).x), sin(2*PI*(ExtendInt(r)).x), 0 ]| by A1
    .= SM.((ExtendInt(r)).x) by Def6
    .= (SM * ExtendInt(r)).x by FUNCT_2:15;
  end;
