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 Th15:
  |(Bn-An,Cn-An)| = |(An-Bn,An-Cn)|
  proof
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
    |(rB-rA,rC-rA)| = |(-(rA-rB),rC-rA)| by RVSUM_1:35
                   .= |(-(rA-rB),-(rA-rC))| by RVSUM_1:35
                   .= |(rA-rB,rA-rC)| by EUCLID_4:25;
    hence thesis;
  end;
