reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;

theorem Th15:
  clf(FX \/ GX) = (clf FX) \/ (clf GX)
proof
  for X be object holds X in clf(FX \/ GX) iff X in (clf FX) \/ (clf GX)
  proof
    let X be object;
A1: now
      assume
A2:   X in (clf FX) \/ (clf GX);
      now
        per cases by A2,XBOOLE_0:def 3;
        suppose
A3:       X in clf FX;
          then reconsider X9= X as Subset of T;
          ex W st X9 = Cl W & W in (FX \/ GX)
          proof
            consider Z such that
A4:         X9 = Cl Z & Z in FX by A3,Def2;
            take Z;
            thus thesis by A4,XBOOLE_0:def 3;
          end;
          hence X in clf(FX \/ GX) by Def2;
        end;
        suppose
A5:       X in clf GX;
          then reconsider X9= X as Subset of T;
          ex W st X9 = Cl W & W in (FX \/ GX)
          proof
            consider Z such that
A6:         X9 = Cl Z & Z in GX by A5,Def2;
            take Z;
            thus thesis by A6,XBOOLE_0:def 3;
          end;
          hence X in clf(FX \/ GX) by Def2;
        end;
      end;
      hence X in clf(FX \/ GX);
    end;
    now
      assume
A7:   X in clf(FX \/ GX);
      then reconsider X9= X as Subset of T;
      consider W such that
A8:   X9 = Cl W and
A9:   W in (FX \/ GX) by A7,Def2;
      now
        per cases by A9,XBOOLE_0:def 3;
        suppose
          W in FX;
          then X9 in clf FX by A8,Def2;
          hence X9 in (clf FX) \/ (clf GX) by XBOOLE_0:def 3;
        end;
        suppose
          W in GX;
          then X9 in clf GX by A8,Def2;
          hence X9 in (clf FX) \/ (clf GX) by XBOOLE_0:def 3;
        end;
      end;
      hence X in (clf FX) \/ (clf GX);
    end;
    hence thesis by A1;
  end;
  hence thesis by TARSKI:2;
end;
