 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;
reserve pn for Point of TOP-REAL n,
        An for Subset of TOP-REAL n,
        Affn for affinely-independent Subset of TOP-REAL n,
        Ak for Subset of TOP-REAL k;

theorem Th10:
  for V be non empty RLSStruct, A be non empty set
  for f be PartFunc of A,the carrier of V
  for X be set holds (r(#)f).:X = r*(f.:X)
proof
  let V be non empty RLSStruct;
  let A be non empty set;
  let f be PartFunc of A,the carrier of V;
  let X be set;
  set rf=r(#)f;
  A1: dom rf=dom f by VFUNCT_1:def 4;
  hereby let y be object;
   assume y in rf.:X;
   then consider x be object such that
    A2: x in dom rf and
    A3: x in X and
    A4: y=rf.x by FUNCT_1:def 6;
   rf.x=rf/.x by A2,PARTFUN1:def 6;
   then A5: y=r*(f/.x) by A2,A4,VFUNCT_1:def 4;
   f.x=f/.x by A1,A2,PARTFUN1:def 6;
   then f/.x in f.:X by A1,A2,A3,FUNCT_1:def 6;
   then y in {r*v where v is Element of V:v in f.:X} by A5;
   hence y in r*(f.:X) by CONVEX1:def 1;
  end;
  let y be object;
  assume y in r*(f.:X);
  then y in {r*v where v is Element of V:v in f.:X} by CONVEX1:def 1;
  then consider v be Element of V such that
   A6: y=r*v and
   A7: v in f.:X;
  consider x be object such that
   A8: x in dom f and
   A9: x in X and
   A10: v=f.x by A7,FUNCT_1:def 6;
  v=f/.x by A8,A10,PARTFUN1:def 6;
  then y=rf/.x by A1,A6,A8,VFUNCT_1:def 4
   .=rf.x by A1,A8,PARTFUN1:def 6;
  hence thesis by A1,A8,A9,FUNCT_1:def 6;
end;
