reserve a for set;
reserve p,p1,p2,q,q1,q2 for Point of TOP-REAL 2;
reserve h1,h2 for FinSequence of TOP-REAL 2;

theorem Th1:
  p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square implies
  ex P1, P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2
  is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
proof
  assume that
A1: p1 <> p2 and
A2: p1 in R^2-unit_square and
A3: p2 in R^2-unit_square;
A4: p1 in L1 \/ L2 or p1 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3;
  per cases by A4,XBOOLE_0:def 3;
  suppose
    p1 in L1;
    hence thesis by A1,A3,Lm30;
  end;
  suppose
    p1 in L2;
    hence thesis by A1,A3,Lm31;
  end;
  suppose
    p1 in L3;
    hence thesis by A1,A3,Lm32;
  end;
  suppose
    p1 in L4;
    hence thesis by A1,A3,Lm33;
  end;
end;
