reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;

theorem Th84:
  for p,q being Point of TOP-REAL 2,r st p`1=q`2 & -p`2=q`1 & p=r
  *q holds p`1=0 & p`2=0 & p=0.TOP-REAL 2
proof
  let p,q be Point of TOP-REAL 2,r;
A1: 1+r*r>0+0 by XREAL_1:8,63;
  assume p`1=q`2 & -p`2=q`1 & p=r*q;
  then
A2: p=|[r*(-p`2),r*(p`1)]| by EUCLID:57;
  then p`2=r*(p`1);
  then p`1=-(r*(r*(p`1))) by A2
    .=-(r*r*(p`1));
  then (1+r*r)*p`1=0;
  hence
A3: p`1=0 by A1,XCMPLX_1:6;
  p`1=r*(-p`2) by A2;
  then p`2=-(r*r*(p`2)) by A2;
  then (1+r*r)*p`2=0;
  hence p`2=0 by A1,XCMPLX_1:6;
  hence thesis by A3,EUCLID:53,54;
end;
