reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th35:
  for X,Y be set, f1,f2 be PartFunc of REAL,REAL-NS n
  st f1|X is bounded & f2|Y is bounded
  holds (f1+f2) | (X /\ Y) is bounded & (f1-f2) | (X /\ Y) is bounded
  proof
    let X,Y  be set,  f1,f2 be PartFunc of REAL,REAL-NS n;
    assume A1: f1|X is bounded & f2|Y is bounded;
    consider r1 be Real such that
A2: for x be set st x in dom (f1|X)
    holds ||. (f1|X)/.x .|| <r1 by A1;
    consider r2 be Real such that
A3: for x be set st x in dom (f2|Y)
    holds ||. (f2|Y)/.x .|| <r2 by A1;
    now let x be set;
      assume x in dom ((f1+f2) | (X /\ Y));
      then
A4:   x in dom (f1+f2) & x in (X /\ Y)  by RELAT_1:57;
      then x in (dom f1) /\ (dom f2)  by VFUNCT_1:def 1;
      then
A5:   x in dom f1 & x in dom f2 by XBOOLE_0:def 4;
A6:   x in X & x in Y by A4,XBOOLE_0:def 4;
      ((f1+f2) | (X /\ Y))/.x = (f1+f2)/.x by A4,PARTFUN2:17
      .= f1/.x + f2/.x by A4,VFUNCT_1:def 1
      .= (f1|X)/.x + f2/.x by A5,A6,PARTFUN2:17
      .= (f1|X)/.x + (f2|Y)/.x by A5,A6,PARTFUN2:17;
      then
A7:   ||. ((f1+f2) | (X /\ Y))/.x .||
      <= ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| by NORMSP_1:def 1;
      x in dom (f1|X) by A5,A6,RELAT_1:57;
      then
A8:   ||.(f1|X)/.x .|| < r1 by A2;
      x in dom (f2|Y) by A5,A6,RELAT_1:57;
      then ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| < r1+r2 by A3,A8,XREAL_1:8;
      hence
      ||. ((f1+f2) | (X /\ Y))/.x .|| < r1+r2 by A7,XXREAL_0:2;
    end;
    hence (f1+f2) | (X /\ Y) is bounded;
    take r1+r2;
    let x be set;
    assume x in dom ((f1-f2) | (X /\ Y));
    then
A9: x in dom (f1-f2) & x in (X /\ Y)  by RELAT_1:57;
    then x in (dom f1) /\ (dom f2)  by VFUNCT_1:def 2;
    then
A10: x in dom f1 & x in dom f2 by XBOOLE_0:def 4;
A11: x in X & x in Y by A9,XBOOLE_0:def 4;
     ((f1-f2) | (X /\ Y))/.x = (f1-f2)/.x by A9,PARTFUN2:17
     .= f1/.x - f2/.x by A9,VFUNCT_1:def 2
     .= (f1|X)/.x - f2/.x by A10,A11,PARTFUN2:17
     .= (f1|X)/.x - (f2|Y)/.x by A10,A11,PARTFUN2:17;
     then
A12: ||. ((f1-f2) | (X /\ Y))/.x .||
     <= ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| by NORMSP_1:3;
     x in dom (f1|X) by A10,A11,RELAT_1:57;
     then
A13: ||.(f1|X)/.x .|| < r1 by A2;
     x in dom (f2|Y) by A10,A11,RELAT_1:57;
     then ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| < r1+r2 by A13,A3,XREAL_1:8;
     hence ||. ((f1-f2) | (X /\ Y))/.x .|| < r1+r2 by A12,XXREAL_0:2;
   end;
