reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th53:
  p1,p2 split P & q1 in P & q2 in P & q1 <> q2 implies q1,q2 split P
proof
  assume that
A1: p1,p2 split P and
A2: q1 in P and
A3: q2 in P and
A4: q1 <> q2;
A5: p2,p1 split P by A1,Th50;
  per cases;
  suppose
    p1 = q1;
    hence thesis by A1,A3,A4,Th51;
  end;
  suppose
    p1 = q2;
    hence thesis by A2,A4,A5,Th52;
  end;
  suppose
    p1 <> q1;
    then p1,q1 split P by A1,A2,Th51;
    then q2,q1 split P by A3,A4,Th52;
    hence thesis by Th50;
  end;
  suppose
    p2 <> q1;
    then q1,p2 split P by A1,A2,Th52;
    hence thesis by A3,A4,Th51;
  end;
end;
