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 Th13:
  |(Bn,Cn)| -|(Cn,Cn)| -|(An,Bn)|+|(An,Cn)| = |(Cn-An,Bn-Cn)| &
  |(Bn-Cn,Bn-Cn)|+|(Cn-An,Bn-Cn)|           = |(Bn-Cn,Bn-An)|
  proof
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
    (rB-rC) + (rC-rA) = (rB-rC+rC)-rA by RVSUM_1:40
                     .= rB-rA by RVSUM_1:43;
    hence thesis by EUCLID_4:28,EUCLID_4:31;
  end;
