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 Th4:
  r(#)L is LinearFunc of F
  proof
    consider g be Point of F such that
A1: for p be Real holds L/.p = p*g by Def2;
A2: r(#)L is total by VFUNCT_1:34;
    now
      let p be Real;
       reconsider pp=p as Element of REAL by XREAL_0:def 1;
      thus (r(#)L)/.p = r*L/.pp by VFUNCT_1:39
      .= r*(p*g) by A1
      .= (r*p)*g by RLVECT_1:def 7
      .= p*(r*g) by RLVECT_1:def 7;
    end;
    hence thesis by A2,Def2;
  end;
