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 & g is_strictly_convex_on X & ( a>0 & b>=0
  or a>=0 & b>0 ) implies a(#)f+b(#)g is_strictly_convex_on X
proof
  assume that
A1: f is_strictly_convex_on X and
A2: g is_strictly_convex_on X and
A3: a>0 & b>=0 or a>=0 & b>0;
  now
    per cases by A3;
    suppose
      a>0 & b>0;
      then
      a(#)f is_strictly_convex_on X & b(#)g is_strictly_convex_on X by A1,A2
,Th12;
      hence thesis by Th13;
    end;
    suppose
A4:   a>0 & b=0;
A5:   X c= dom g by A2;
      a(#)f is_strictly_convex_on X by A1,A4,Th12;
      hence thesis by A4,A5,Th14,RFUNCT_3:57;
    end;
    suppose
A6:   a=0 & b>0;
A7:   X c= dom f by A1;
      b(#)g is_strictly_convex_on X by A2,A6,Th12;
      hence thesis by A6,A7,Th14,RFUNCT_3:57;
    end;
  end;
  hence thesis;
end;
