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
  A <> B & B <> C & |(B-A,B-C)| = 0 implies angle(A,B,C)=PI/2 or
  angle(A,B,C)=3/2*PI
  proof
    assume that
A1: A <> B and
A2: B <> C and
A3: |(B-A,B-C)| = 0;
    |(A-B,C-B)| = |( -(A-B), -(C-B) )| by EUCLID_2:23
               .= |( B-A, -(C-B) )| by RVSUM_1:35
               .= 0 by A3,RVSUM_1:35;
    hence thesis by A1,A2,EUCLID_3:45;
  end;
