reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem Th9:
  An <> Bn & An <> Cn & |(An-Bn,An-Cn)| = 0 & L1 = Line(An,Bn) &
  L2 = Line(An,Cn) implies L1 _|_ L2
  proof
    assume that
A1: An <> Bn and
A2: An <> Cn and
A3: |(An-Bn,An-Cn)| = 0 and
A4: L1 = Line(An,Bn) and
A5: L2 = Line(An,Cn);
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
A6: L1 = Line(rA,rB) by A4,EUCLID12:4;
A7: L2 = Line(rA,rC) by A5,EUCLID12:4;
A8: (rA-rB)<>0*n by A1,EUCLIDLP:9;
A9: (rA-rC)<>0*n by A2,EUCLIDLP:9;
    (rA-rB) _|_ (rA-rC) by A8,A9,EUCLIDLP:def 3,A3,RVSUM_1:def 17;
    hence thesis by A6,A7,EUCLIDLP:def 8;
  end;
