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 Th3:
  r(#)L is LinearFunc
proof
  consider g such that
A1: for p holds L.p = g*p by Def3;
A2: L is total by Def3;
  now
    let p;
     reconsider pp = p as Element of REAL by XREAL_0:def 1;
    thus (r(#)L).p = r*L.pp by A2,RFUNCT_1:57
      .= r*(g*p) by A1
      .= r*g*p;
  end;
  hence thesis by A2,Def3;
end;
