reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  for a,b being Function of Y,BOOLEAN,PA being a_partition of Y
  holds B_SUP(a 'or' b,PA)=B_SUP(a,PA) 'or' B_SUP(b,PA)
proof
  let a,b be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
    let y be Element of Y;
A1: now
      assume
A2:   ex x being Element of Y st x in EqClass(y,PA) & (a 'or' b).x= TRUE;
      then consider x1 being Element of Y such that
A3:   x1 in EqClass(y,PA) and
A4:   (a 'or' b).x1=TRUE;
A5:   a.x1 = FALSE & b.x1 = FALSE or a.x1 = FALSE & b.x1 = TRUE or a.x1 =
      TRUE & b.x1 = FALSE or a.x1 = TRUE & b.x1 = TRUE by XBOOLEAN:def 3;
A6:  a.x1 'or' b.x1=TRUE by A4,Def4;
      now
        per cases by A6,A5;
        case
          a.x1 = FALSE & b.x1 = TRUE;
          then (B_SUP(b,PA)).y = TRUE by A3,Def17;
          then (B_SUP(a,PA)).y 'or' (B_SUP(b,PA)).y = TRUE;
          then (B_SUP(a,PA) 'or' B_SUP(b,PA)).y = TRUE by Def4;
          hence thesis by A2,Def17;
        end;
        case
          a.x1 = TRUE & b.x1 = FALSE;
          then (B_SUP(a,PA)).y = TRUE by A3,Def17;
          then (B_SUP(a,PA)).y 'or' (B_SUP(b,PA)).y = TRUE;
          then (B_SUP(a,PA) 'or' B_SUP(b,PA)).y = TRUE by Def4;
          hence thesis by A2,Def17;
        end;
        case
          a.x1 = TRUE & b.x1 = TRUE;
          then (B_SUP(b,PA)).y = TRUE by A3,Def17;
          then (B_SUP(a,PA)).y 'or' (B_SUP(b,PA)).y = TRUE;
          then (B_SUP(a,PA) 'or' B_SUP(b,PA)).y = TRUE by Def4;
          hence thesis by A2,Def17;
        end;
      end;
      hence thesis;
    end;
    now
      assume
A7:  not (ex x being Element of Y st x in EqClass(y,PA) & (a 'or' b) .x=TRUE);
      now
        assume ex x being Element of Y st x in EqClass(y,PA) & b.x=TRUE;
        then consider x1 being Element of Y such that
A8:    x1 in EqClass(y,PA) and
A9:    b.x1=TRUE;
        a.x1 'or' b.x1 = TRUE by A9;
        then (a 'or' b).x1 = TRUE by Def4;
        hence contradiction by A7,A8;
      end;
      then
A10:  (B_SUP(b,PA)).y = FALSE by Def17;
      now
        assume ex x being Element of Y st x in EqClass(y,PA) & a.x=TRUE;
        then consider x1 being Element of Y such that
A11:    x1 in EqClass(y,PA) and
A12:    a.x1=TRUE;
        a.x1 'or' b.x1 = TRUE by A12;
        then (a 'or' b).x1 = TRUE by Def4;
        hence contradiction by A7,A11;
      end;
      then (B_SUP(a,PA)).y 'or' (B_SUP(b,PA)).y = FALSE 'or' FALSE by A10,Def17
;
      then (B_SUP(a,PA) 'or' B_SUP(b,PA)).y = FALSE by Def4;
      hence thesis by A7,Def17;
    end;
    hence thesis by A1;
end;
