reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for F be PartFunc of REAL,REAL, X be set, r st 0<r holds F
  is_convex_on X iff r(#)F is_convex_on X
proof
  let F be PartFunc of REAL,REAL, X be set, r;
  assume
A1: 0<r;
A2: dom F = dom(r(#)F) by VALUED_1:def 5;
  thus F is_convex_on X implies r(#)F is_convex_on X
  proof
    assume
A3: F is_convex_on X;
    then
A4: X c= dom F;
    thus X c= dom(r(#)F) by A2,A3;
    let p be Real;
    assume
A5: 0<=p & p<=1;
    let x,y be Real;
    assume that
A6: x in X and
A7: y in X and
A8: p*x+(1-p)*y in X;
    F.(p*x+(1-p)*y)<=p*F.x + (1-p)*F.y by A3,A5,A6,A7,A8;
    then r* F.(p*x+(1-p)*y)<=r*(p*F.x + (1-p)*F.y) by A1,XREAL_1:64;
    then (r(#)F).(p*x+(1-p)*y)<=p*(r*F.x) + (1-p)*r*F.y by A2,A4,A8,
VALUED_1:def 5;
    then (r(#)F).(p*x+(1-p)*y)<=p*(r(#)F).x + (1-p)*(r*F.y) by A2,A4,A6,
VALUED_1:def 5;
    hence thesis by A2,A4,A7,VALUED_1:def 5;
  end;
  assume
A9: r(#)F is_convex_on X;
  then
A10: X c= dom(r(#)F);
  hence X c= dom F by VALUED_1:def 5;
  let p be Real;
  assume
A11: 0<=p & p<=1;
  let x,y be Real;
  assume that
A12: x in X and
A13: y in X and
A14: p*x+(1-p)*y in X;
  (r(#)F).(p*x+(1-p)*y)<=p*(r(#)F).x + (1-p)*(r(#)F).y by A9,A11,A12,A13,A14;
  then r*F.(p*x+(1-p)*y)<=p*(r(#)F).x + (1-p)*(r(#)F).y by A10,A14,
VALUED_1:def 5;
  then r*F.(p*x+(1-p)*y)<=p*(r*F.x) + (1-p)*(r(#)F).y by A10,A12,VALUED_1:def 5
;
  then r*F.(p*x+(1-p)*y)<=p*(r*F.x) + (1-p)*(r*F.y) by A10,A13,VALUED_1:def 5;
  then r*F.(p*x+(1-p)*y)/r <= r*(p*F.x + (1-p)*F.y)/r by A1,XREAL_1:72;
  then F.(p*x+(1-p)*y) <= r*(p*F.x + (1-p)*F.y)/r by A1,XCMPLX_1:89;
  hence thesis by A1,XCMPLX_1:89;
end;
