reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;
reserve L,L0,L1,L2 for Element of line_of_REAL n;
reserve P,P0,P1,P2 for Element of plane_of_REAL n;

theorem Th109:
  L1,L2 are_coplane & L1 _|_ L2 implies L1 meets L2
proof
  assume
A1: L1,L2 are_coplane;
  assume
A2: L1 _|_ L2;
  then
A3: L2 is being_line by Th67;
  L1 is being_line by A2,Th67;
  then consider x0 such that
A4: x0 in L1 and
A5: not x0 in L2 by A2,A3,Th75,Th79;
  consider L such that
A6: x0 in L and
A7: L _|_ L2 and
A8: L meets L2 by A3,A5,Th62;
  consider x such that
A9: x in L and
A10: x in L2 by A8,Th49;
  x in L1
  proof
A11: L1 meets L by A4,A6,Th49;
    consider P such that
    P is being_plane and
A12: L c= P & L2 c= P by A7,A8,Th107;
    consider P0 such that
A13: L1 c= P0 & L2 c= P0 by A1,Th96;
A14: P = P0 by A3,A4,A5,A6,A13,A12,Th102;
    consider L0 such that
A15: x in L0 and
A16: L0 _|_ L2 and
A17: L0 // L1 by A2,Th80;
    assume
A18: not x in L1;
    then consider P1 such that
A19: L0 c= P1 and
A20: L1 c= P1 and
    P1 is being_plane by A15,A17,Th106;
    L1 is being_line by A17,Th66;
    then P = P1 by A10,A18,A13,A14,A15,A19,A20,Th102;
    then L = L0 by A7,A9,A10,A12,A15,A16,A19,Th108;
    hence contradiction by A18,A15,A17,A11,Th71;
  end;
  hence thesis by A10,Th49;
end;
