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