reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem Th114:
  X1,X2 are_weakly_separated implies for g being Function of X1
  union X2,Y holds g is continuous Function of X1 union X2,Y iff g|X1 is
  continuous Function of X1,Y & g|X2 is continuous Function of X2,Y
proof
  assume
A1: X1,X2 are_weakly_separated;
  let g be Function of X1 union X2,Y;
A2: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A3: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  hence g is continuous Function of X1 union X2,Y implies g|X1 is continuous
  Function of X1,Y & g|X2 is continuous Function of X2,Y by A2,Th82;
  thus g|X1 is continuous Function of X1,Y & g|X2 is continuous Function of X2
  ,Y implies g is continuous Function of X1 union X2,Y
  proof
    assume that
A4: g|X1 is continuous Function of X1,Y and
A5: g|X2 is continuous Function of X2,Y;
    for x being Point of X1 union X2 holds g is_continuous_at x
    proof
      set X0 = X1 union X2;
      let x be Point of X1 union X2;
A6:   X1 meets X2 implies g is_continuous_at x
      proof
        assume
A7:     X1 meets X2;
A8:     now
          assume
A9:       ( not X1 is SubSpace of X2)& not X2 is SubSpace of X1;
          then consider Y1, Y2 being open non empty SubSpace of X such that
A10:      Y1 meet X0 is SubSpace of X1 and
A11:      Y2 meet X0 is SubSpace of X2 and
A12:      X0 is SubSpace of Y1 union Y2 or ex Z being closed non
empty SubSpace of X st the TopStruct of X = (Y1 union Y2) union Z & Z meet X0
          is SubSpace of X1 meet X2 by A1,A7,TSEP_1:89;
A13:      Y2 meets X0 implies Y2 meet X0 is open SubSpace of X0 by Th39;
A14:      Y1 meets X0 implies Y1 meet X0 is open SubSpace of X0 by Th39;
A15:      now
            X is SubSpace of X by TSEP_1:2;
            then reconsider X12 = the TopStruct of X as SubSpace of X by Th6;
            assume
A16:        not X0 is SubSpace of Y1 union Y2;
            then consider Z being closed non empty SubSpace of X such that
A17:        the TopStruct of X = (Y1 union Y2) union Z and
A18:        Z meet X0 is SubSpace of X1 meet X2 by A12;
            the carrier of X0 c= the carrier of X12 by BORSUK_1:1;
            then
A19:        X0 is SubSpace of X12 by TSEP_1:4;
            then X12 meets X0 by Th17;
            then
A20:        ((Y1 union Y2) union Z) meet X0 = the TopStruct of X0 by A17,A19,
TSEP_1:28;
A21:        Y1 meets X0 & Y2 meets X0 by A7,A9,A10,A11,A17,A18,Th32;
A22:        now
A23:          now
                given x2 being Point of Y2 meet X0 such that
A24:            x2 = x;
                g|(Y2 meet X0) is continuous by A2,A5,A11,Th83;
                then g|(Y2 meet X0) is_continuous_at x2;
                hence thesis by A7,A9,A10,A11,A13,A17,A18,A24,Th32,Th79;
              end;
A25:          now
                given x1 being Point of Y1 meet X0 such that
A26:            x1 = x;
                g|(Y1 meet X0) is continuous by A3,A4,A10,Th83;
                then g|(Y1 meet X0) is_continuous_at x1;
                hence thesis by A7,A9,A10,A11,A14,A17,A18,A26,Th32,Th79;
              end;
              assume
A27:          ex x12 being Point of (Y1 union Y2) meet X0 st x12 = x;
              (Y1 union Y2) meet X0 = (Y1 meet X0) union (Y2 meet X0) by A21,
TSEP_1:32;
              hence thesis by A27,A25,A23,Th11;
            end;
A28:        now
              given x0 being Point of Z meet X0 such that
A29:          x0 = x;
              consider x00 being Point of X1 meet X2 such that
A30:          x00 = x0 by A18,Th10;
              consider x1 being Point of X1 such that
A31:          x1 = x00 by A7,Th12;
              consider x2 being Point of X2 such that
A32:          x2 = x00 by A7,Th12;
              g|X1 is_continuous_at x1 & g|X2 is_continuous_at x2 by A4,A5,Th44
;
              hence thesis by A29,A30,A31,A32,Th111;
            end;
            (Y1 union Y2) meets X0 & Z meets X0 by A7,A9,A10,A11,A16,A17,A18
,Th33;
            then ((Y1 union Y2) meet X0) union (Z meet X0) = the TopStruct of
            X0 by A20,TSEP_1:32;
            hence thesis by A22,A28,Th11;
          end;
          now
            assume
A33:        X0 is SubSpace of Y1 union Y2;
            then
A34:        Y1 meets X0 by A9,A10,A11,Th31;
A35:        now
              given x2 being Point of Y2 meet X0 such that
A36:          x2 = x;
              g|(Y2 meet X0) is continuous by A2,A5,A11,Th83;
              then g|(Y2 meet X0) is_continuous_at x2;
              hence thesis by A9,A10,A11,A13,A33,A36,Th31,Th79;
            end;
A37:        now
              given x1 being Point of Y1 meet X0 such that
A38:          x1 = x;
              g|(Y1 meet X0) is continuous by A3,A4,A10,Th83;
              then g|(Y1 meet X0) is_continuous_at x1;
              hence thesis by A9,A10,A11,A14,A33,A38,Th31,Th79;
            end;
            Y1 is SubSpace of Y1 union Y2 by TSEP_1:22;
            then Y1 union Y2 meets X0 by A34,Th18;
            then
A39:        (Y1 union Y2) meet X0 = X0 by A33,TSEP_1:28;
            Y2 meets X0 by A9,A10,A11,A33,Th31;
            then (Y1 meet X0) union (Y2 meet X0) = X0 by A34,A39,TSEP_1:32;
            hence thesis by A37,A35,Th11;
          end;
          hence thesis by A15;
        end;
        now
A40:      now
            assume X2 is SubSpace of X1;
            then
A41:        the TopStruct of X1 = X0 by TSEP_1:23;
            then reconsider x1 = x as Point of X1;
            g|X1 is_continuous_at x1 by A4,Th44;
            hence thesis by A41,Th81;
          end;
A42:      now
            assume X1 is SubSpace of X2;
            then
A43:        the TopStruct of X2 = X0 by TSEP_1:23;
            then reconsider x2 = x as Point of X2;
            g|X2 is_continuous_at x2 by A5,Th44;
            hence thesis by A43,Th81;
          end;
          assume X1 is SubSpace of X2 or X2 is SubSpace of X1;
          hence thesis by A42,A40;
        end;
        hence thesis by A8;
      end;
      X1 misses X2 implies g is_continuous_at x
      proof
        assume X1 misses X2;
        then X1,X2 are_separated by A1,TSEP_1:78;
        then consider Y1, Y2 being open non empty SubSpace of X such that
A44:    X1 is SubSpace of Y1 and
A45:    X2 is SubSpace of Y2 and
A46:    Y1 misses Y2 or Y1 meet Y2 misses X0 by TSEP_1:77;
        Y2 misses X1 by A44,A45,A46,Th30;
        then
A47:    X2 is open SubSpace of X0 by A45,Th41;
A48:    now
          given x2 being Point of X2 such that
A49:      x2 = x;
          g|X2 is_continuous_at x2 by A5,Th44;
          hence thesis by A47,A49,Th79;
        end;
        Y1 misses X2 by A44,A45,A46,Th30;
        then
A50:    X1 is open SubSpace of X0 by A44,Th41;
        now
          given x1 being Point of X1 such that
A51:      x1 = x;
          g|X1 is_continuous_at x1 by A4,Th44;
          hence thesis by A50,A51,Th79;
        end;
        hence thesis by A48,Th11;
      end;
      hence thesis by A6;
    end;
    hence thesis by Th44;
  end;
end;
