reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];
reserve a for Element of [:Fin X, Fin X:];
reserve A for set;
reserve x,y for Element of [:Fin X, Fin X:],
  a,b for Element of DISJOINT_PAIRS X;
reserve A for set,
  x for Element of [:Fin A, Fin A:],
  a,b,c,d,s,t for Element of DISJOINT_PAIRS A,
  B,C,D for Element of Fin DISJOINT_PAIRS A;
reserve K,L,M for Element of Normal_forms_on A;

theorem Th43:
  mi (B \/ C) c= mi B \/ C
proof
  now
    let a;
    assume
A1: a in mi(B \/ C);
    then
A2: a in B \/ C by Th36;
    now
      per cases by A2,XBOOLE_0:def 3;
      suppose
A3:     a in B;
        now
          let b;
          assume b in B;
          then b in B \/ C by XBOOLE_0:def 3;
          hence b c= a implies b = a by A1,Th36;
        end;
        then a in mi B by A3,Th39;
        hence a in mi B \/ C by XBOOLE_0:def 3;
      end;
      suppose
        a in C;
        hence a in mi B \/ C by XBOOLE_0:def 3;
      end;
    end;
    hence a in mi B \/ C;
  end;
  hence thesis by Lm5;
end;
