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 Th107:
  L1 _|_ L2 & L1 meets L2 implies ex P st P is being_plane & L1 c= P & L2 c= P
proof
  assume that
A1: L1 _|_ L2 and
A2: L1 meets L2;
  consider x1 such that
A3: x1 in L1 and
A4: x1 in L2 by A2,Th49;
  L1 is being_line by A1,Th67;
  then consider x2 such that
A5: x2 <> x1 & x2 in L1 by Th53;
A6: L1 = Line(x1,x2) by A3,A5,Th64;
  L2 is being_line by A1,Th67;
  then consider x3 such that
A7: x3 <> x1 & x3 in L2 by Th53;
  reconsider P = plane(x1,x2,x3) as Subset of REAL n;
  take P;
A8: x1 in P & x2 in P by Th82;
A9: x3 in P by Th82;
A10: L2 = Line(x1,x3) by A4,A7,Th64;
  x2 - x1,x3 - x1 are_lindependent2 by A1,A3,A4,A5,A7,Th45,Th74;
  hence thesis by A6,A10,A8,A9,Th85,Th90;
end;
