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;
