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 Th75:
  for L1,L2 st L1 _|_ L2 holds L1 <> L2
proof
  let L1,L2;
  assume
A1: L1 _|_ L2;
  now
    per cases;
    case
A2:   L1 misses L2;
      ex x st x in L1 by Th52;
      hence thesis by A2,Th49;
    end;
    case
      L1 meets L2;
      then consider x0 such that
A3:   x0 in L1 and
A4:   x0 in L2 by Th49;
      L1 is being_line by A1,Th67;
      then consider x1 such that
A5:   x1 <> x0 and
A6:   x1 in L1 by Th53;
A7:   x1 - x0 <> 0*n by A5,Th9;
      L2 is being_line by A1,Th67;
      then consider x2 such that
A8:   x2 <> x0 and
A9:   x2 in L2 by Th53;
A10:  x2 - x0 <> 0*n by A8,Th9;
A11:  x1 - x0 _|_ x2 - x0 by A1,A3,A4,A5,A6,A8,A9,Th74;
      not x1 in L2
      proof
        assume x1 in L2;
        then ex a st x1 - x0 = a * (x2 - x0) by A4,A8,A9,Th70;
        then x1 - x0 // x2 - x0 by A7,A10;
        hence contradiction by A11,Lm3,Th45;
      end;
      hence thesis by A6;
    end;
  end;
  hence thesis;
end;
