reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;

theorem Th1935:
  for X,Y be set, f1,f2 be PartFunc of REAL,the carrier of Z
  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,the carrier of Z;
   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 A41: 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 & x in X & x in Y by A41,XBOOLE_0:def 4; then
A6:  x in dom(f2|Y) by RELAT_1:57;
    ((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,PARTFUN2:17
      .= (f1|X)/.x + (f2|Y)/.x by A5,PARTFUN2:17; then
A7:   ||. ((f1+f2) | (X /\ Y))/.x .||
      <= ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| by NORMSP_1:def 1;
    ||.(f1|X)/.x .|| < r1 by A2,A5,RELAT_1:57; then
    ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| < r1+r2 by A3,A6,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
A8: 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 & x in X & x in Y by A8,XBOOLE_0:def 4; then
A11:x in dom (f2|Y) by RELAT_1:57;
   ((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,PARTFUN2:17
     .= (f1|X)/.x - (f2|Y)/.x by A10,PARTFUN2:17; then
A12: ||. ((f1-f2) | (X /\ Y))/.x .||
     <= ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| by NORMSP_1:3;
   ||.(f1|X)/.x .|| < r1 by A2,A10,RELAT_1:57;
   then ||.(f1|X)/.x .|| + ||.(f2|Y)/.x .|| < r1+r2 by A11,A3,XREAL_1:8;
   hence ||. ((f1-f2) | (X /\ Y))/.x .|| < r1+r2 by A12,XXREAL_0:2;
end;
