reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem
  for a,b being set holds <*a,b*> in y>=0-plane iff a in REAL & ex y st
  b = y & y >= 0
proof
  let a,b be set;
  hereby
    assume <*a,b*> in y>=0-plane;
    then consider x,y such that
A2: <*a,b*> = |[x,y]| and
A3: y >= 0;
    <*a,b*>.1 = x by A2,FINSEQ_1:44;
    hence a in REAL by XREAL_0:def 1;
    take y;
    <*a,b*>.2 = b;
    hence b = y & y >= 0 by A3,A2,FINSEQ_1:44;
  end;
  assume a in REAL;
  then reconsider x = a as Real;
  given y such that
A4: b = y and
A5: y >= 0;
  |[x,y]| = <*a,b*> by A4;
  hence thesis by A5;
end;
