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-line iff a in REAL & b = 0
proof
  let a,b be set;
A1: <*a,b*> in y=0-line iff ex x st <*a,b*> = |[x,0]|;
  hereby
    assume <*a,b*> in y=0-line;
    then consider x,y such that
A3: <*a,b*> = |[x,0]| by A1;
    <*a,b*>.1 = x by A3,FINSEQ_1:44;
    hence a in REAL by XREAL_0:def 1;
    <*a,b*>.2 = b;
    hence b = 0 by A3,FINSEQ_1:44;
  end;
  assume a in REAL;
  then reconsider x = a as Real;
  |[x,0]| = <*a,0 *>;
  hence thesis;
end;
