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 Th112:
  L0 c= P & L1 c= P & L2 c= P & L0 // L1 & L1 // L2 & L0 <> L1 &
  L meets L0 & L meets L1 implies L meets L2
proof
  assume that
A1: L0 c= P & L1 c= P and
A2: L2 c= P and
A3: L0 // L1 and
A4: L1 // L2 and
A5: L0 <> L1;
A6: L1 is being_line by A3,Th66;
  assume that
A7: L meets L0 and
A8: L meets L1;
  consider x0 such that
A9: x0 in L and
A10: x0 in L0 by A7,Th49;
A11: L0 misses L1 by A3,A5,Th71;
  then not x0 in L1 by A10,Th49;
  then consider L9 being Element of line_of_REAL n such that
A12: x0 in L9 and
A13: L9 _|_ L1 and
A14: L9 meets L1 by A6,Th62;
  consider y1 such that
A15: y1 in L9 and
A16: y1 in L1 by A14,Th49;
A17: x0 <> y1 by A10,A11,A16,Th49;
  then
A18: L9 = Line(x0,y1) by A12,A15,Th64;
  then L9 c= P by A1,A10,A16,Th95;
  then L9,L2 are_coplane by A2,Th96;
  then
A19: L9 meets L2 by A4,A13,Th61,Th109;
  then consider y2 such that
A20: y2 in L9 and
A21: y2 in L2 by Th49;
  consider a such that
A22: y2 - x0 = a*(y1 - x0) by A12,A15,A17,A20,Th70;
  L2 is being_line by A4,Th66;
  then consider x2 such that
A23: x2 <> y2 & x2 in L2 by Th53;
  consider x1 such that
A24: x1 in L and
A25: x1 in L1 by A8,Th49;
  x0 <> x1 by A10,A25,A11,Th49;
  then
A26: L = Line(x0,x1) by A9,A24,Th64;
A27: L2 = Line(y2,x2) by A21,A23,Th64;
  now
    per cases;
    case
      x1 = y1;
      hence thesis by A9,A24,A17,A18,A19,Th64;
    end;
    case
A28:  x1 <> y1;
      set x = (1 - a)*x0 + a*x1;
      consider b such that
A29:  b <> 0 and
A30:  x2 - y2 = b*(x1 - y1) by A4,A25,A16,A21,A23,A28,Th32,Th77;
A31:  x1 - y1 = 1 * (x1 - y1) by EUCLID_4:3
        .= (1/b*b)*(x1 - y1) by A29,XCMPLX_1:87
        .= 1/b*(x2 - y2) by A30,EUCLID_4:4;
      x = 1 * x0 + -a*x0 + a*x1 by Th11
        .= x0 + -a*x0 + a*x1 by EUCLID_4:3
        .= (a*x1 + -a*x0) + x0 by RVSUM_1:15
        .= a*(x1 - x0) + x0 by Th12
        .= a*(x1 + 0*n + -x0) + x0 by EUCLID_4:1
        .= a*(x1 + (-y1 + y1) + -x0) + x0 by Th2
        .= a*(x1 + -y1 + y1 + -x0) + x0 by RVSUM_1:15
        .= a*(x1 - y1 + (y1 + -x0)) + x0 by RVSUM_1:15
        .= a*((1/b)*(x2 - y2)) + a*(y1 - x0) + x0 by A31,EUCLID_4:6
        .= (a*(1/b))*(x2 - y2) + a*(y1 - x0) + x0 by EUCLID_4:4
        .= (a/b)*(x2 - y2) + a*(y1 - x0) + x0 by XCMPLX_1:99
        .= (a/b)*(x2 - y2) + (y2 + -x0 + x0) by A22,RVSUM_1:15
        .= (a/b)*(x2 - y2) + (y2 + (-x0 + x0)) by RVSUM_1:15
        .= (a/b)*(x2 - y2) + (y2 + 0*n) by Th2
        .= (a/b)*(x2 - y2) + y2 by EUCLID_4:1
        .= (a/b)*x2 + -(a/b)*y2 + y2 by Th12
        .= y2 + -(a/b)*y2 + (a/b)*x2 by RVSUM_1:15
        .= 1 * y2 + -(a/b)*y2 + (a/b)*x2 by EUCLID_4:3
        .= (1 - a/b)* y2 + (a/b)*x2 by Th11;
      then
A32:  x in L2 by A27;
      x in L by A26;
      hence thesis by A32,Th49;
    end;
  end;
  hence thesis;
end;
