reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;
reserve p,q for Point of TOP-REAL n;
reserve f,f1,f2 for homogeneous additive Function of TOP-REAL n,TOP-REAL n;

theorem Th33:
  for f be rotation Function of TOP-REAL n,TOP-REAL n st
      f is X-support-yielding &
      for i st i in X/\Seg n holds p.i = 0
    holds f.p = p
proof
  set TR=TOP-REAL n;
  let f be rotation Function of TR,TR such that
  A1: f is X-support-yielding and
  A2: for i st i in X/\Seg n holds p.i=0;
  set sp=sqr p,sfp=sqr(f.p);
  A3: Sum sp>=0 by RVSUM_1:86;
  Sum sfp>=0 & |.f.p.|=|.p.| by Def4,RVSUM_1:86;
  then A4: Sum sp=Sum sfp by A3,SQUARE_1:28;
  A5: len p=n by CARD_1:def 7; then
  A6: len sp=n by RVSUM_1:143;
  A7: len(f.p)=n by CARD_1:def 7; then
  len sfp=n by RVSUM_1:143;
  then reconsider sp,sfp as Element of n-tuples_on REAL by A6,FINSEQ_2:92;
  A8: dom f=the carrier of TR by FUNCT_2:52;
  A9: for i st i in Seg n holds sp.i<=sfp.i
  proof
    let i;
    A10: sp.i=(p.i)^2 & sfp.i=(f.p.i)^2 by VALUED_1:11;
    assume A11: i in Seg n;
    per cases;
    suppose i in X;
      then i in X/\Seg n by A11,XBOOLE_0:def 4;
      then p.i=0 by A2;
      hence thesis by A10,XREAL_1:63;
    end;
    suppose not i in X;
      hence thesis by A1,A8,A10;
    end;
  end;
  for i st 1<=i & i<=n holds p.i=f.p.i
  proof
    let i;
    A12: sp.i=(p.i)^2 by VALUED_1:11;
    assume 1<=i & i<=n;
    then A13: i in Seg n;
    then A14: sp.i>=sfp.i & sp.i<=sfp.i by A4,A9,RVSUM_1:83;
    per cases;
    suppose i in X;
      then A15: i in X/\Seg n by A13,XBOOLE_0:def 4;
      then p.i=0 by A2;
      then (f.p.i)^2=0 by A12,A14,VALUED_1:11;
      then f.p.i=0;
      hence thesis by A2,A15;
    end;
    suppose not i in X;
      hence thesis by A1,A8;
    end;
  end;
  hence thesis by A5,A7;
end;
