reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  X misses Y implies chi(X,C) + chi(Y,C) = chi(X \/ Y,C)
proof
  assume
A1: X /\ Y = {};
A2: now
    let c such that
A3: c in dom (chi(X,C) + chi(Y,C));
    now
      per cases;
      suppose
A4:     c in X;
        then not c in Y by A1,XBOOLE_0:def 4;
        then
A5:     chi(Y,C).c = 0 by Th61;
A6:     c in X \/ Y by A4,XBOOLE_0:def 3;
        chi(X,C).c = 1 by A4,Th61;
        hence chi(X,C).c + chi(Y,C).c = chi(X \/ Y,C).c by A5,A6,Th61;
      end;
      suppose
A7:     not c in X;
        then
A8:     chi(X,C).c = 0 by Th61;
        now
          per cases;
          suppose
A9:         c in Y;
            then
A10:        c in X \/ Y by XBOOLE_0:def 3;
            chi(Y,C).c = 1 by A9,Th61;
            hence chi(X,C).c + chi(Y,C).c = chi(X \/ Y,C).c by A8,A10,Th61;
          end;
          suppose
A11:        not c in Y;
            then
A12:        not c in X \/ Y by A7,XBOOLE_0:def 3;
            chi(Y,C).c = 0 by A11,Th61;
            hence chi(X,C).c + chi(Y,C).c = chi(X \/ Y,C).c by A8,A12,Th61;
          end;
        end;
        hence chi(X,C).c + chi(Y,C).c = chi(X \/ Y,C).c;
      end;
    end;
    hence (chi(X,C) + chi(Y,C)).c = chi(X \/ Y,C).c by A3,VALUED_1:def 1;
  end;
  dom (chi(X,C) + chi(Y,C)) = dom chi(X,C) /\ dom chi(Y,C) by VALUED_1:def 1
    .= C /\ dom chi(Y,C) by Th61
    .= C /\ C by Th61
    .= dom chi(X \/ Y,C) by Th61;
  hence thesis by A2,PARTFUN1:5;
end;
