reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k for Nat;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th45:
  f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & (for
  r st r<x0 ex g st r<g & g<x0 & g in dom(f1+f2)) implies f1+f2
  is_left_convergent_in x0 & lim_left(f1+f2,x0)=lim_left(f1,x0)+lim_left(f2,x0)
proof
  assume that
A1: f1 is_left_convergent_in x0 and
A2: f2 is_left_convergent_in x0 and
A3: for r st r<x0 ex g st r<g & g<x0 & g in dom(f1+f2);
A4: now
    let seq;
    assume that
A5: seq is convergent and
A6: lim seq=x0 and
A7: rng seq c=dom(f1+f2)/\left_open_halfline(x0);
A8: dom(f1+f2)=dom f1/\dom f2 by A7,Lm4;
A9: rng seq c=dom f1/\left_open_halfline(x0) by A7,Lm4;
A10: rng seq c=dom f2/\left_open_halfline(x0) by A7,Lm4;
    then
A11: lim(f2/*seq)=lim_left(f2,x0) by A2,A5,A6,Def7;
A12: f2/*seq is convergent by A2,A5,A6,A10;
    rng seq c=dom(f1+f2) by A7,Lm4;
    then
A13: f1/*seq+f2/*seq=(f1+f2)/*seq by A8,RFUNCT_2:8;
A14: f1/*seq is convergent by A1,A5,A6,A9;
    hence (f1+f2)/*seq is convergent by A12,A13;
    lim(f1/*seq)=lim_left(f1,x0) by A1,A5,A6,A9,Def7;
    hence lim((f1+f2)/*seq)=lim_left(f1,x0)+lim_left(f2,x0) by A14,A12,A11,A13,
SEQ_2:6;
  end;
  hence f1+f2 is_left_convergent_in x0 by A3;
  hence thesis by A4,Def7;
end;
