reserve a,b,p,r,r1,r2,s,s1,s2,x0,x for Real;
reserve f,g for PartFunc of REAL,REAL;
reserve X,Y for set;

theorem
  f is_strictly_convex_on X iff f-r is_strictly_convex_on X
proof
A1: dom(f-r) = dom f by VALUED_1:3;
A2: for x being Element of REAL st x in dom (f-r) holds ((f-r)-(-r)).x = f.x
  proof
    let x be Element of REAL;
    assume
A3: x in dom (f-r);
    then ((f-r)-(-r)).x=(f-r).x-(-r) by VALUED_1:3
      .=(f-r).x+r
      .=f.x-r+r by A1,A3,VALUED_1:3
      .= f.x-(r-r);
    hence thesis;
  end;
  dom((f-r)-(-r))=dom(f-r) by VALUED_1:3;
  then (f-r)-(-r) = f by A1,A2,PARTFUN1:5;
  hence thesis by Lm1;
end;
