reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p 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 seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem Th3:
  L1+L2 is LinearFunc of F & L1-L2 is LinearFunc of F
  proof
    consider g1 be Point of F such that
A1: for p be Real holds L1/.p = p*g1 by Def2;
    consider g2 be Point of F such that
A2: for p be Real holds L2/.p = p*g2 by Def2;
A3: L1+L2 is total by VFUNCT_1:32;
    now
      let p be Real;
       reconsider pp=p as Element of REAL by XREAL_0:def 1;
      thus (L1+L2)/.p = L1/.pp + L2/.pp by VFUNCT_1:37
      .= p*g1 + L2/.pp by A1
      .= p*g1 + p*g2 by A2
      .= p*(g1+g2) by RLVECT_1:def 5;
    end;
    hence L1+L2 is LinearFunc of F by A3,Def2;
A4: L1-L2 is total by VFUNCT_1:32;
    now
      let p be Real;
       reconsider pp=p as Element of REAL by XREAL_0:def 1;
      thus (L1-L2)/.p = L1/.pp - L2/.pp by VFUNCT_1:37
      .= p*g1 - L2/.pp by A1
      .= p*g1 - p*g2 by A2
      .= p*(g1-g2) by RLVECT_1:34;
    end;
    hence thesis by A4,Def2;
  end;
