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 X c= dom f & for a,b,r st a in X & b in
X & r in X & a < r & r < b holds (f.r-f.a)/(r-a) < (f.b-f.a)/(b-a) & (f.b-f.a)/
  (b-a) < (f.b-f.r)/(b-r)
proof
A1: X c= dom f & (for a,b,r st a in X & b in X & r in X & a < r & r < b
  holds (f.r-f.a)/(r-a) < (f.b-f.a)/(b-a) & (f.b-f.a)/(b-a) < (f.b-f.r)/(b-r))
  implies f is_strictly_convex_on X
  proof
    assume that
A2: X c= dom f and
A3: for a,b,r st a in X & b in X & r in X & a<r & r<b holds (f.r-f.a)
    / (r-a) < (f.b-f.a)/(b-a) & (f.b-f.a)/(b-a) < (f.b-f.r)/(b-r);
    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.(p*r + (1-p)*s) < p*f.r + (1-p)*f.s
    proof
      let p be Real;
      assume that
A4:   0<p and
A5:   p<1;
A6:   1-p > 0 by A5,XREAL_1:50;
      for s,t being Real
         st s in X & t in X & p*s+(1-p)*t in X & s <> t
      holds f.(p*s + (1-p)*t) < p*f.s + (1-p)*f.t
      proof
        let s,t be Real;
        assume that
A7:     s in X & t in X & p*s+(1-p)*t in X and
A8:     s <> t;
        now
          per cases by A8,XXREAL_0:1;
          suppose
            s<t;
            then
A9:         t-s > 0 by XREAL_1:50;
            (p*s+(1-p)*t)-s =(1-p)*(t-s);
            then
A10:        (p*s+(1-p)*t)-s > 0 by A6,A9,XREAL_1:129;
            then
A11:        (p*s+(1-p)*t) > s by XREAL_1:47;
A12:        (f.(p*s+(1-p)*t)-f.s)*(t-s)*p =p*((t-s)*f.(p*s+(1-p)*t))-p*((
t-s)*f.s) & (f. t-f.(p*s+(1-p)*t))*(t-s)*(1-p) =(1-p)*((t-s)*f.t)-(1-p)*((t-s)*
            f.(p*s+(1-p)*t));
            t-(p*s+(1-p)*t) =p*(t-s);
            then
A13:        t-(p*s+(1-p)*t) > 0 by A4,A9,XREAL_1:129;
            then (p*s+(1-p)*t) < t by XREAL_1:47;
            then
            (f.(p*s+(1-p)*t)-f.s)/((p*s+(1-p)*t)-s) < (f.t-f.s)/(t-s) & (
f.t-f.s)/(t-s)< (f.t-f.(p*s+(1-p)*t))/(t-(p*s+(1-p)*t)) by A3,A7,A11;
            then
            (f.(p*s+(1-p)*t)-f.s)/((p*s+(1-p)*t)-s) < (f.t-f.(p*s+(1-p)*t
            ))/(t-(p*s+(1-p)*t)) by XXREAL_0:2;
            then
            (f.(p*s+(1-p)*t)-f.s)*(t-(p*s+(1-p)*t)) < (f.t-f.(p*s+(1-p)*t
            ))*((p*s+(1-p)*t)-s) by A13,A10,XREAL_1:102;
            then
            p*((t-s)*f.(p*s+(1-p)*t))+(1-p)*((t-s)*f.(p*s+(1-p)*t)) <(1-p
            )*((t-s)*f.t)+p*((t-s)*f.s) by A12,XREAL_1:21;
            then f.(p*s+(1-p)*t) <((1-p)*f.t*(t-s)+p*f.s*(t-s))/(t-s) by A9,
XREAL_1:81;
            then f.(p*s+(1-p)*t) <((1-p)*f.t*(t-s))/(t-s)+(p*f.s*(t-s))/(t-s)
            by XCMPLX_1:62;
            then f.(p*s+(1-p)*t) <(1-p)*f.t+(p*f.s*(t-s))/(t-s) by A9,
XCMPLX_1:89;
            hence thesis by A9,XCMPLX_1:89;
          end;
          suppose
            s>t;
            then
A14:        s-t > 0 by XREAL_1:50;
            (p*s+(1-p)*t)-t =p*(s-t);
            then
A15:        (p*s+(1-p)*t)-t > 0 by A4,A14,XREAL_1:129;
            then
A16:        (p*s+(1-p)*t) > t by XREAL_1:47;
A17:        (f.(p*s+(1-p)*t)-f.t)*(s-t)*(1-p) =(1-p)*((s-t)*f.(p*s+(1-p)*
t))-(1-p)*((s-t) *f.t) & (f.s-f.(p*s+(1-p)*t))*(s-t)*p =p*((s-t)*f.s)-p*((s-t)*
            f.(p*s+(1-p)*t));
            s-(p*s+(1-p)*t) =(1-p)*(s-t);
            then
A18:        s-(p*s+(1-p)*t) > 0 by A6,A14,XREAL_1:129;
            then (p*s+(1-p)*t) < s by XREAL_1:47;
            then
            (f.(p*s+(1-p)*t)-f.t)/((p*s+(1-p)*t)-t) < (f.s-f.t)/(s-t) & (
f.s-f.t)/(s-t)< (f.s-f.(p*s+(1-p)*t))/(s-(p*s+(1-p)*t)) by A3,A7,A16;
            then
            (f.(p*s+(1-p)*t)-f.t)/((p*s+(1-p)*t)-t) < (f.s-f.(p*s+(1-p)*t
            ))/(s-(p*s+(1-p)*t)) by XXREAL_0:2;
            then
            (f.(p*s+(1-p)*t)-f.t)*(s-(p*s+(1-p)*t)) < (f.s-f.(p*s+(1-p)*t
            ))*((p*s+(1-p)*t)-t) by A18,A15,XREAL_1:102;
            then
            (1-p)*((s-t)*f.(p*s+(1-p)*t))+p*((s-t)*f.(p*s+(1-p)*t)) <p*((
            s-t)*f.s)+(1-p)*((s-t)*f.t) by A17,XREAL_1:21;
            then f.(p*s+(1-p)*t) <(p*f.s*(s-t)+(1-p)*f.t*(s-t))/(s-t) by A14,
XREAL_1:81;
            then f.(p*s+(1-p)*t) <(p*f.s*(s-t))/(s-t)+((1-p)*f.t*(s-t))/(s-t)
            by XCMPLX_1:62;
            then f.(p*s+(1-p)*t) <p*f.s+((1-p)*f.t*(s-t))/(s-t) by A14,
XCMPLX_1:89;
            hence thesis by A14,XCMPLX_1:89;
          end;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis by A2;
  end;
  f is_strictly_convex_on X implies X c= dom f & for a,b,r st a in X & b
in X & r in X & a < r & r < b holds (f.r-f.a)/(r-a) < (f.b-f.a)/(b-a) & (f.b-f.
  a)/(b-a) < (f.b-f.r)/(b-r)
  proof
    assume
A19: f is_strictly_convex_on X;
    for a,b,r st a in X & b in X & r in X & a < r & r < b holds (f.r-f.a)/
    (r-a) < (f.b-f.a)/(b-a) & (f.b-f.a)/(b-a) < (f.b-f.r)/(b-r)
    proof
      let a,b,r;
      assume that
A20:  a in X & b in X & r in X and
A21:  a < r and
A22:  r < b;
      reconsider p = (b-r)/(b-a) as Real;
      reconsider aa=a, bb=b as Real;
A23:  b-r < b-a & 0 < b-r by A21,A22,XREAL_1:10,50;
A24:  p+(r-a)/(b-a) = ((b-r)+(r-a))/(b-a) by XCMPLX_1:62
        .= 1 by A23,XCMPLX_1:60;
      then p*a + (1-p)*b = (a*(b-r))/(b-a)+b*((r-a)/(b-a)) by XCMPLX_1:74
        .= (a*(b-r))/(b-a)+(b*(r-a))/(b-a) by XCMPLX_1:74
        .= ((b*a-r*a)+(r-a)*b)/(b-a) by XCMPLX_1:62
        .= r*(b-a)/(b-a);
      then
A25:  p*a + (1-p)*b = r by A23,XCMPLX_1:89;
      0 < (b-r)/(b-a) & (bb-r)/(bb-aa) < 1 by A23,XREAL_1:139,189;
      then
A26:  f.r < p*f.a + (1-p)*f.b by A19,A20,A21,A22,A25;
A27:  0 < r-a by A21,XREAL_1:50;
A28:  (p*f.a+(1-p)*f.b)*(b-a)= (b-a)*p*f.a+(b-a)*((1-p)*f.b)
        .= ((b-a)*(b-r)/(b-a))*f.a + ((b-a)*((r-a)/(b-a))*f.b) by A24,
XCMPLX_1:74
        .= ((b-r)*(b-a))/(b-a)*f.a + ((b-a)*(r-a)/(b-a))*f.b by XCMPLX_1:74
        .= (b-r)*((b-a)/(b-a))*f.a + ((b-a)*(r-a)/(b-a))*f.b by XCMPLX_1:74
        .= (b-r)*1*f.a + ((r-a)*(b-a))/(b-a)*f.b by A23,XCMPLX_1:60
        .= (b-r)*f.a + (r-a)*((b-a)/(b-a))*f.b by XCMPLX_1:74
        .= (b-r)*f.a + (r-a)*1*f.b by A23,XCMPLX_1:60;
      then f.r*(b-a) < (b-a)*f.a +((r-a)*f.b-(r-a)*f.a) by A23,A26,XREAL_1:68;
      then f.r*(b-a)-(b-a)*f.a < (r-a)*f.b-(r-a)*f.a by XREAL_1:19;
      then (b-a)*(f.r-f.a) < (r-a)*(f.b-f.a);
      hence (f.r-f.a)/(r-a) < (f.b-f.a)/(b-a) by A23,A27,XREAL_1:106;
      f.r*(b-a) < (r-b)*f.b + (b-r)*f.a + (b-a)*f.b by A23,A26,A28,XREAL_1:68;
      then f.r*(b-a)-((r-b)*f.b + (b-r)*f.a) < (b-a)*f.b by XREAL_1:19;
      then f.r*(b-a)+(-((r-b)*f.b+(b-r)*f.a))<(b-a)*f.b;
      then -(r-b)*f.b -(b-r)*f.a < (b-a)*f.b - (b-a)*f.r by XREAL_1:20;
      then (b-r)*(f.b-f.a) < (b-a)*(f.b-f.r);
      hence thesis by A23,XREAL_1:106;
    end;
    hence thesis by A19;
  end;
 hence thesis by A1;
end;
