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,Y be set st F is_convex_on X & Y c=
  X holds F is_convex_on Y
proof
  let F be PartFunc of REAL,REAL, X,Y be set;
  assume that
A1: F is_convex_on X and
A2: Y c= X;
  X c= dom F by A1;
  hence Y c= dom F by A2;
  let p be Real;
  assume
A3: 0<=p & p<=1;
  let x,y be Real;
  assume x in Y & y in Y & p*x+(1-p)*y in Y;
  hence thesis by A1,A2,A3;
end;
