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 st X c= dom F holds 0(#) F
  is_convex_on X
proof
  let F be PartFunc of REAL,REAL, X be set;
  assume
A1: X c= dom F;
  hence X c= dom(0(#)F) by VALUED_1:def 5;
  let p be Real;
  assume that
  0<=p and
  p<=1;
  let x,y be Real;
  assume that
A2: x in X and
A3: y in X and
A4: p*x + (1-p)*y in X;
A5: dom F = dom(0(#)F) by VALUED_1:def 5;
  then
A6: (0(#)F).(p*x+(1-p)*y) = 0* F.(p*x+(1-p)*y) by A1,A4,VALUED_1:def 5
    .= 0;
  p*(0(#)F).x + (1-p)*(0(#)F).y = p*(0* F.x) + (1-p)*(0(#)F).y by A1,A5,A2,
VALUED_1:def 5
    .= p*0 + (1-p)*(0* F.y) by A1,A5,A3,VALUED_1:def 5
    .= 0 + 0;
  hence thesis by A6;
end;
