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 Th14:
  f is_strictly_convex_on X & g is_convex_on X implies f+g
  is_strictly_convex_on X
proof
  assume
A1: f is_strictly_convex_on X & g is_convex_on X;
  then X c= dom f & X c= dom g by RFUNCT_3:def 12;
  then X c= dom f /\ dom g by XBOOLE_1:19;
  then
A2: X c= dom (f+g) by VALUED_1:def 1;
  for p being Real st 0<p & p<1
 for r,s being Real st r in X & s in
X & p*r + (1-p)*s in X & r <> s holds (f+g).(p*r + (1-p)*s) < p*(f+g).r + (1-p)
  *(f+g).s
  proof
    let p be Real;
    assume
A3: 0<p & p<1;
    for r,s being Real
   st r in X & s in X & p*r + (1-p)*s in X & r <> s
    holds (f+g).(p*r + (1-p)*s) < p*(f+g).r + (1-p)*(f+g).s
    proof
      let r,s be Real;
      assume that
A4:   r in X and
A5:   s in X and
A6:   p*r + (1-p)*s in X and
A7:   r <> s;
A8:   (f+g).(p*r + (1-p)*s) = f.(p*r+(1-p)*s)+g.(p*r+(1-p)*s) & (f+g).r =
      (f.r+g.r ) by A2,A4,A6,VALUED_1:def 1;
A9:   (p*f.r+(1-p)*f.s)+(p*g.r+(1-p)*g.s) = p*(f.r+g.r) + (1-p)*(f.s+g.s)
      & (f+g). s = (f.s+g.s) by A2,A5,VALUED_1:def 1;
      f.(p*r+(1-p)*s) < p*f.r+(1-p)*f.s & g.(p*r+(1-p)*s) <= p*g.r+(1-p)*
      g.s by A1,A3,A4,A5,A6,A7,RFUNCT_3:def 12;
      hence thesis by A8,A9,XREAL_1:8;
    end;
    hence thesis;
  end;
  hence thesis by A2;
end;
