reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem Th35:
  L1 is being_line implies ex L2 st x in L2 & L1 _|_ L2
  proof
    assume
A1: L1 is being_line;
    per cases;
    suppose x in L1;
      consider x1 be Element of REAL 2  such that
A2:   not x1 in L1 by Th15;
      consider L2 such that
      x1 in L2 and
A3:   L1 _|_ L2 and
      L1 meets L2 by A1,A2,EUCLIDLP:62;
      consider L3 such that
A4:   x in L3 and
A5:   L3 _|_ L1 and
      L3 // L2 by A3,EUCLIDLP:80;
      thus thesis by A4,A5;
    end;
    suppose not x in L1;
      then consider L2 such that
A6:   x in L2 and
A7:   L1 _|_ L2 and
      L1 meets L2 by A1,EUCLIDLP:62;
      thus thesis by A6,A7;
    end;
  end;
