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_strongly_quasiconvex_on X holds f is_quasiconvex_on X
proof
  let X,f such that
A1: f is_strongly_quasiconvex_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 holds f.(p*r + (1-p)*s) <= max(f.r,f.s)
  proof
    let p be Real such that
A3: 0<p & p<1;
    for r,s being Real st r in X & s in X & p*r + (1-p)*s in X holds f.(p*
    r + (1-p)*s) <= max(f.r,f.s)
    proof
      let r,s be Real such that
A4:   r in X & s in X & p*r + (1-p)*s in X;
      now
        per cases;
        suppose
          r<>s;
          hence thesis by A1,A3,A4;
        end;
        suppose
          r=s;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
  X c= dom f by A1;
  hence thesis by A2;
end;
