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
  for X, f st f is_strictly_convex_on X holds f is_strongly_quasiconvex_on X
proof
  let X, f such that
A1: f is_strictly_convex_on X;
A2: for p being Real st 0<p & p<1 holds for r,s being Real st r in X & s in
  X & p*r + (1-p)*s in X & r <> s holds f.(p*r + (1-p)*s) < max(f.r,f.s)
  proof
    let p be Real;
    assume that
A3: 0<p and
A4: 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.(p*r + (1-p)*s) < max(f.r,f.s)
    proof
      let r,s be Real;
      1-p > 0 by A4,XREAL_1:50;
      then
A5:   (1-p)*f.s<=(1-p)*max(f.r,f.s) by XREAL_1:64,XXREAL_0:25;
      assume r in X & s in X & p*r + (1-p)*s in X & r <> s;
      then
A6:   f.(p*r + (1-p)*s) < p*f.r + (1-p)*f.s by A1,A3,A4;
      p*f.r<=p*max(f.r,f.s) by A3,XREAL_1:64,XXREAL_0:25;
      then p*f.r + (1-p)*f.s <= p*max(f.r,f.s)+(1-p)*max(f.r,f.s) by A5,
XREAL_1:7;
      hence thesis by A6,XXREAL_0:2;
    end;
    hence thesis;
  end;
  X c= dom f by A1;
  hence thesis by A2;
end;
