reserve y for object, X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1 for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve h for non-zero 0-convergent Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem Th2:
  L1+L2 is LinearFunc & L1-L2 is LinearFunc
proof
  consider g1 such that
A1: for p holds L1.p = g1*p by Def3;
  consider g2 such that
A2: for p holds L2.p = g2*p by Def3;
A3: L1 is total & L2 is total by Def3;
  now
    let p;
     reconsider pp=p as Element of REAL by XREAL_0:def 1;
    thus (L1+L2).p = L1.pp + L2.pp by A3,RFUNCT_1:56
      .= g1*p + L2.p by A1
      .= g1*p + g2*p by A2
      .= (g1+g2)*p;
  end;
  hence L1+L2 is LinearFunc by A3,Def3;
  now
    let p;
     reconsider pp=p as Element of REAL by XREAL_0:def 1;
    thus (L1-L2).p = L1.pp - L2.pp by A3,RFUNCT_1:56
      .= g1*p - L2.p by A1
      .= g1*p - g2*p by A2
      .= (g1-g2)*p;
  end;
  hence thesis by A3,Def3;
end;
