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 Th110:
  L1 c= P & L2 c= P & L1 _|_ L2 & x in P & L0 // L2 & x in L0
  implies L0 c= P & L0 _|_ L1
proof
  assume that
A1: L1 c= P & L2 c= P and
A2: L1 _|_ L2 and
A3: x in P;
  L1,L2 are_coplane by A1,Th96;
  then L1 meets L2 by A2,Th109;
  then consider x0 such that
A4: x0 in L1 and
A5: x0 in L2 by Th49;
  L2 is being_line by A2,Th67;
  then consider x1 such that
A6: x1 <> x0 and
A7: x1 in L2 by Th53;
A8: plane(x,x1,x0) c= P by A1,A3,A4,A7,Th91;
  assume that
A9: L0 // L2 and
A10: x in L0;
  L0 is being_line by A9,Th66;
  then consider x2 such that
A11: x2 <> x & x2 in L0 by Th53;
  consider a such that
  a <> 0 and
A12: x2 - x = a*(x1 - x0) by A9,A10,A5,A6,A7,A11,Th32,Th77;
A13: 1 + a + (-a) = 1;
  x2 = x + a*(x1 - x0) by A12,Th6
    .= 1 * x + a*(x1 - x0) by EUCLID_4:3
    .= 1 * x + (a * x1 + (-a) * x0) by Th12
    .= 1 * x + a * x1 + (-a) * x0 by RVSUM_1:15;
  then
A14: x2 in plane(x,x1,x0) by A13;
  L0 = Line(x2,x) by A10,A11,Th64;
  hence thesis by A2,A3,A9,A14,A8,Th61,Th95;
end;
