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
  x1 in P & x2 in P & y1 in P & y2 in P & x2 - x1, y2 - y1
  are_lindependent2 implies Line(x1,x2) meets Line(y1,y2)
proof
  reconsider L1 = Line(x1,x2), L2 = Line(y1,y2) as Element of line_of_REAL n
  by Th47;
  assume that
A1: x1 in P & x2 in P & y1 in P & y2 in P and
A2: x2 - x1, y2 - y1 are_lindependent2;
A3: x1 in L1 & x2 in L1 by EUCLID_4:9;
  L1 c= P & L2 c= P by A1,Th95;
  then
A4: L1,L2 are_coplane by Th96;
A5: y1 in L2 & y2 in L2 by EUCLID_4:9;
  y2 - y1 <> 0*n by A2,Lm2;
  then
A6: y2 <> y1 by Th9;
  then
A7: L2 is being_line;
  x2 - x1 <> 0*n by A2,Lm2;
  then
A8: x2 <> x1 by Th9;
  then
A9: L1 is being_line;
  L1 meets L2
  proof
    assume L1 misses L2;
    then L1 // L2 by A4,A9,A7,Th113;
    hence contradiction by A2,A8,A6,A3,A5,Lm3,Th77;
  end;
  hence thesis;
end;
