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 Th113:
  L1,L2 are_coplane & L1 is being_line & L2 is being_line & L1
  misses L2 implies L1 // L2
proof
  assume that
A1: L1,L2 are_coplane and
A2: L1 is being_line and
A3: L2 is being_line;
  assume
A4: L1 misses L2;
  then consider x such that
A5: x in L1 and
A6: not x in L2 by Th63;
  consider P such that
A7: L1 c= P and
A8: L2 c= P by A1,Th96;
  consider L9 being Element of line_of_REAL n such that
A9: x in L9 and
A10: L9 _|_ L2 and
A11: L9 meets L2 by A3,A6,Th62;
  consider x2 such that
A12: x2 in L9 and
A13: x2 in L2 by A11,Th49;
  consider L0 such that
A14: x in L0 and
A15: L0 _|_ L9 and
A16: L0 // L2 by A10,Th80;
  L9 = Line(x2,x) by A6,A9,A12,A13,Th64;
  then
A17: L9 c= P by A5,A7,A8,A13,Th95;
  then
A18: L0 c= P by A8,A9,A10,A14,A16,Th110;
A19: L9 is being_line by A15,Th67;
  consider x1 such that
A20: x1 <> x and
A21: x1 in L1 by A2,Th53;
A22: L1 = Line(x,x1) by A5,A20,A21,Th64;
A23: L0 meets L1 by A5,A14,Th49;
  L1 = L0
  proof
A24: not x1 in L9 by A4,A9,A11,A20,A22,Th64;
    then consider L such that
A25: x1 in L and
A26: L9 _|_ L and
A27: L9 meets L by A19,Th62;
A28: L meets L1 by A21,A25,Th49;
    assume L1 <> L0;
    then
A29: L <> L0 by A14,A20,A22,A25,Th64;
    consider x9 being Element of REAL n such that
A30: x9 in L9 and
A31: x9 in L by A27,Th49;
    L = Line(x9,x1) by A24,A25,A30,A31,Th64;
    then
A32: L c= P by A7,A17,A21,A30,Th95;
    then L // L0 by A17,A15,A18,A26,Th111;
    hence contradiction by A4,A8,A16,A18,A23,A32,A29,A28,Th112;
  end;
  hence thesis by A16;
end;
