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;

theorem Th71:
  L1 // L2 & L1<>L2 implies L1 misses L2
proof
  assume that
A1: L1 // L2 and
A2: L1<>L2;
  assume not L1 misses L2;
  then consider x be object such that
A3: x in L1 and
A4: x in L2 by XBOOLE_0:3;
  reconsider x as Element of REAL n by A3;
  consider x1, x2, y1, y2 being Element of REAL n such that
A5: L1 = Line(x1,x2) and
A6: L2 = Line(y1,y2) and
A7: (x2 - x1) // (y2 - y1) by A1;
A8: x2 - x1 <> 0*n by A7;
  then x1 <> x2 by Th9;
  then
A9: L1 is being_line by A5;
  then consider x0 such that
A10: x <> x0 and
A11: x0 in L1 by EUCLID_4:14;
A12: x0 - x <> 0*n by A10,Th9;
  ex a st x0 - x = a*(x2 - x1) by A5,A3,A11,Th31;
  then x0 - x // x2 - x1 by A8,A12;
  then
A13: x0 - x // y2 - y1 by A7,Th33;
A14: y2 - y1 <> 0*n by A7;
  then y1 <> y2 by Th9;
  then
A15: L2 is being_line by A6;
  then consider y0 such that
A16: x <> y0 and
A17: y0 in L2 by EUCLID_4:14;
A18: y0 - x <> 0*n by A16,Th9;
  ex b st y0 - x = b*(y2 - y1) by A6,A4,A17,Th31;
  then y0 - x // y2 - y1 by A14,A18;
  then
A19: x0 - x // y0 - x by A13,Th33;
A20: Line(x,x0) c= Line(x,y0)
  proof
    let y be object;
    assume y in Line(x,x0);
    then consider t such that
A21: y = (1-t)*x + t*x0;
    consider a such that
A22: x0 - x = a*(y0 - x) by A19;
    y = 1 * x + -t*x + t*x0 by A21,Th11
      .= x + -t*x + t*x0 by EUCLID_4:3
      .= x + (t*x0 + -t*x) by RVSUM_1:15
      .= x + t*(x0 - x) by Th12;
    then y = x + (t*a)*(y0 - x) by A22,EUCLID_4:4
      .= x + ((t*a)*y0 + -(t*a)*x) by Th12
      .= x + -(t*a)*x + (t*a)*y0 by RVSUM_1:15
      .= 1 * x + -(t*a)*x + (t*a)*y0 by EUCLID_4:3
      .= (1 -(t*a))*x + (t*a)*y0 by Th11;
    hence thesis;
  end;
A23: Line(x,y0) c= Line(x,x0)
  proof
    let y be object;
    assume y in Line(x,y0);
    then consider t such that
A24: y = (1-t)*x + t*y0;
    consider a such that
A25: y0 - x = a*(x0 - x) by A19,Def1;
    y = 1 * x + -t*x + t*y0 by A24,Th11
      .= x + -t*x + t*y0 by EUCLID_4:3
      .= x + (t*y0 + -t*x) by RVSUM_1:15
      .= x + t*(y0 - x) by Th12;
    then y = x + (t*a)*(x0 - x) by A25,EUCLID_4:4
      .= x + ((t*a)*x0 + -(t*a)*x) by Th12
      .= x + -(t*a)*x + (t*a)*x0 by RVSUM_1:15
      .= 1 * x + -(t*a)*x + (t*a)*x0 by EUCLID_4:3
      .= (1 -(t*a))*x + (t*a)*x0 by Th11;
    hence thesis;
  end;
A26: L2 = Line(x,y0) by A4,A15,A16,A17,Th30;
  L1 = Line(x,x0) by A3,A9,A10,A11,Th30;
  hence contradiction by A2,A26,A20,A23,XBOOLE_0:def 10;
end;
