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 Th108:
  L0 c= P & L1 c= P & L2 c= P & x in L0 & x in L1 & x in L2 & L0
  _|_ L2 & L1 _|_ L2 implies L0 = L1
proof
  assume that
A1: L0 c= P and
A2: L1 c= P and
A3: L2 c= P and
A4: x in L0 and
A5: x in L1 and
A6: x in L2;
A7: L1 meets L0 by A4,A5,Th49;
  assume that
A8: L0 _|_ L2 and
A9: L1 _|_ L2;
  consider x0 such that
A10: x <> x0 and
A11: x0 in L0 and
  not x0 in L2 by A6,A8,Th81;
  L1 is being_line by A9,Th67;
  then consider x1 such that
A12: x1 <> x and
A13: x1 in L1 by Th53;
  consider x2 such that
A14: x <> x2 and
A15: x2 in L2 and
  not x2 in L1 by A5,A9,Th81;
A16: x0 - x _|_ x2 - x by A4,A6,A8,A10,A11,A14,A15,Th74;
  then P = plane(x,x0,x2) by A1,A3,A4,A11,A15,Th45,Th92;
  then consider a1,a2,a3 such that
A17: a1 + a2 + a3 = 1 & x1 = a1 * x + a2 * x0 + a3 * x2 by A2,A13,Th88;
  x0 - x,x2 - x are_orthogonal by A16;
  then
A18: |(x0 - x,x2 - x)|=0 by RVSUM_1:def 17;
A19: x1 - x = -x + (x + a2*(x0 - x) + a3*(x2 - x)) by A17,Th27
    .= -x + (x + (a2*(x0 - x) + a3*(x2 - x))) by RVSUM_1:15
    .= (-x + x) + (a2*(x0 - x) + a3*(x2 - x)) by RVSUM_1:15
    .= 0*n + (a2*(x0 - x) + a3*(x2 - x)) by Th2
    .= a2*(x0 - x) + a3*(x2 - x) by EUCLID_4:1;
  x2 - x <> 0*n by A14,Th9;
  then
A20: |(x2 - x,x2 - x)| <> 0 by EUCLID_4:17;
  x1 - x _|_ x2 - x by A5,A6,A9,A14,A15,A12,A13,Th74;
  then x1 - x,x2 - x are_orthogonal;
  then 0 = |(a2*(x0 - x) + a3*(x2 - x),x2 - x)| by A19,RVSUM_1:def 17
    .= |(a2*(x0 - x),x2 - x)| + |(a3*(x2 - x),x2 - x)| by EUCLID_4:20
    .= a2*|(x0 - x,x2 - x)| + |(a3*(x2 - x),x2 - x)| by EUCLID_4:21
    .= a3*|(x2 - x,x2 - x)| by A18,EUCLID_4:21;
  then
A21: x1 - x = a2*(x0 - x) + 0 * (x2 - x) by A19,A20,XCMPLX_1:6
    .= a2*(x0 - x) + 0*n by EUCLID_4:3
    .= a2*(x0 - x) by EUCLID_4:1;
A22: x0 - x <> 0*n by A10,Th9;
  x1 - x <> 0*n by A12,Th9;
  then
A23: x1 - x // x0 - x by A21,A22;
A24: L1 = Line(x,x1) by A5,A12,A13,Th64;
  L0 = Line(x,x0) by A4,A10,A11,Th64;
  then L1 // L0 by A24,A23;
  hence thesis by A7,Th71;
end;
