reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;
reserve f,f1,f2,h for FinSequence of TOP-REAL 2;

theorem
  ex P1, P2 being non empty Subset of TOP-REAL 2 st P1 is being_S-P_arc
& P2 is being_S-P_arc & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {|[ 0,0 ]|, |[
  1,1 ]|}
proof
  consider f1,f2 such that
A1: f1 is being_S-Seq and
A2: f2 is being_S-Seq and
A3: R^2-unit_square = L~f1 \/ L~f2 and
A4: L~f1 /\ L~f2 = {|[ 0,0 ]|, |[ 1,1 ]|} and
  f1/.1=|[0,0]| and
  f1/.len f1=|[1,1]| and
  f2/.1=|[0,0]| and
  f2/.len f2=|[1,1]| by Th24;
  reconsider P1 = L~f1, P2 = L~f2 as non empty Subset of TOP-REAL 2 by A4;
  take P1, P2;
  thus thesis by A1,A2,A3,A4;
end;
