reserve h,h1,h2 for 0-convergent non-zero Real_Sequence,
  c,c1 for constant Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x0,r,r0,r1,r2,g,g1,g2 for Real,
  n0,k,n,m for Element of NAT,
  a,b,d for Real_Sequence,
  x for set;

theorem
  f1 is_right_differentiable_in x0 & f2 is_right_differentiable_in x0
  implies f1+f2 is_right_differentiable_in x0 & Rdiff(f1+f2,x0) = Rdiff(f1,x0)+
  Rdiff(f2,x0)
proof
  assume that
A1: f1 is_right_differentiable_in x0 and
A2: f2 is_right_differentiable_in x0;
  consider r2 such that
A3: r2>0 and
A4: [.x0,x0+r2.] c=dom f2 by A2;
  consider r1 such that
A5: r1>0 and
A6: [.x0,x0+r1.] c= dom f1 by A1;
A7: x0+0 = x0;
  set r = min (r1,r2);
  0 <= r by A5,A3,XXREAL_0:15;
  then
A8: x0 <= x0 + r by A7,XREAL_1:7;
  r<=r2 by XXREAL_0:17;
  then
A9: x0 + r <= x0 + r2 by XREAL_1:7;
  then x0 + r in {g: x0 <= g & g <= x0 +r2} by A8;
  then
A10: x0 + r in [.x0,x0+r2.] by RCOMP_1:def 1;
  x0 <= x0 + r2 by A8,A9,XXREAL_0:2;
  then x0 in [.x0,x0+r2.] by XXREAL_1:1;
  then [.x0,x0+r.] c= [.x0,x0+r2.] by A10,XXREAL_2:def 12;
  then
A11: [.x0,x0+r.] c= dom f2 by A4;
  r<=r1 by XXREAL_0:17;
  then
A12: x0 + r <= x0 + r1 by XREAL_1:7;
  then x0 + r in {g: x0 <= g & g <= x0 +r1} by A8;
  then
A13: x0 + r in [.x0,x0+r1.] by RCOMP_1:def 1;
  x0 <= x0 + r1 by A12,A8,XXREAL_0:2;
  then x0 in [.x0,x0+r1.] by XXREAL_1:1;
  then [.x0,x0+r.] c= [.x0,x0+r1.] by A13,XXREAL_2:def 12;
  then [.x0,x0+r.] c= dom f1 by A6;
  then
A14: [.x0,x0+r.] c= dom f1 /\ dom f2 by A11,XBOOLE_1:19;
  then
A15: [.x0,x0+r.] c= dom(f1 + f2) by VALUED_1:def 1;
A16: for h,c st rng c = {x0} & rng (h+c) c= dom (f1+f2) &
 (for n being Nat holds h.n >
0 ) holds h"(#)((f1+f2)/*(h+c) - (f1+f2)/*c) is convergent & lim (h"(#)((f1+f2)
  /*(h+c) - (f1+f2)/*c)) = Rdiff(f1,x0) + Rdiff(f2,x0)
  proof
    let h,c;
    assume that
A17: rng c = {x0} and
A18: rng (h+c) c= dom (f1+f2) and
A19: for n being Nat holds h.n > 0;
A20: rng (h+c) c= dom f1 /\ dom f2 by A18,VALUED_1:def 1;
A21: now
      let n;
A22:  rng c c= dom f1 /\ dom f2
      proof
        let x be object;
        assume x in rng c;
        then
A23:    x = x0 by A17,TARSKI:def 1;
        x0 in [.x0,x0 + r.] by A8,XXREAL_1:1;
        hence thesis by A14,A23;
      end;
      thus (f1/*(h+c) - f1/*c + (f2/*(h+c) - f2/*c)).n = (f1/*(h+c) + (-(f1/*c
      ))).n + (f2/*(h+c) - f2/*c).n by SEQ_1:7
        .= (f1/*(h+c)).n + (-(f1/*c)).n + (f2/*(h+c) + (-(f2/*c))).n by SEQ_1:7
        .= (f1/*(h+c)).n + (-(f1/*c)).n + ((f2/*(h+c)).n + (-(f2/*c)).n) by
SEQ_1:7
        .= (f1/*(h+c)).n + (f2/*(h+c)).n+ ((-(f1/*c)).n +(-(f2/*c)).n)
        .= (f1/*(h+c)).n + (f2/*(h+c)).n + (-(f1/*c).n + (-(f2/*c)).n) by
SEQ_1:10
        .= (f1/*(h+c)).n + (f2/*(h+c)).n + (-(f1/*c).n + -(f2/*c).n) by
SEQ_1:10
        .= (f1/*(h+c)).n + (f2/*(h+c)).n - ((f1/*c).n + (f2/*c).n)
        .= (f1/*(h+c) + f2/*(h+c)).n - ((f1/*c).n + (f2/*c).n) by SEQ_1:7
        .= (f1/*(h+c) + f2/*(h+c)).n - ((f1/*c + f2/*c).n) by SEQ_1:7
        .= (f1/*(h+c) + f2/*(h+c) - (f1/*c + f2/*c)).n by RFUNCT_2:1
        .= ((f1+f2)/*(h+c) - (f1/*c + f2/*c)).n by A20,RFUNCT_2:8
        .= ((f1+f2)/*(h+c) - (f1+f2)/*c).n by A22,RFUNCT_2:8;
    end;
    then
A24: f1/*(h+c)-f1/*c + (f2/*(h+c) - f2/*c)=(f1+f2)/*(h+c) - (f1+f2)/*c by
FUNCT_2:63;
    dom f1 /\dom f2 c= dom f2 by XBOOLE_1:17;
    then
A25: rng (h+c) c= dom f2 by A20;
    then
A26: lim(h"(#)(f2/*(h+c) - f2/*c)) = Rdiff(f2,x0) by A2,A17,A19,Th15;
A27: h"(#)(f2/*(h+c) - f2/*c) is convergent by A2,A17,A19,A25;
    dom f1 /\ dom f2 c= dom f1 by XBOOLE_1:17;
    then
A28: rng (h+c) c= dom f1 by A20;
A29: (h"(#)(f1/*(h+c) - f1/*c) +h"(#)(f2/*(h+c) - f2/*c)) = h"(#)(f1/*(h+c
    ) -f1/*c + (f2/*(h+c) - f2/*c)) by SEQ_1:16;
A30: h"(#)(f1/*(h+c) - f1/*c) is convergent by A1,A17,A19,A28;
    then (h"(#)(f1/*(h+c) - f1/*c) + h"(#)(f2/*(h+c) - f2/*c)) is convergent
    by A27;
    hence h"(#)((f1+f2)/*(h+c) - (f1+f2)/*c) is convergent by A29,A21,
FUNCT_2:63;
    lim(h"(#)(f1/*(h+c) - f1/*c)) = Rdiff(f1,x0) by A1,A17,A19,A28,Th15;
    hence thesis by A30,A27,A26,A29,A24,SEQ_2:6;
  end;
  0<r by A5,A3,XXREAL_0:15;
  hence thesis by A15,A16,Th15;
end;
