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 Th12:
  0<r implies (f is_strictly_convex_on X iff r(#)f is_strictly_convex_on X)
proof
A1: dom((1/r)(#)(r(#)f))=dom(r(#)f) by VALUED_1:def 5;
  assume
A2: 0<r;
A3: for x being Element of REAL st x in dom(r(#)f) holds ((1/r)(#)(r(#)f)).x
  = f.x
  proof
    let x be Element of REAL;
    assume
A4: x in dom (r(#)f);
    then ((1/r)(#)(r(#)f)).x = (1/r)*(r(#)f).x by A1,VALUED_1:def 5
      .= (1/r)*(r*f.x) by A4,VALUED_1:def 5
      .= (1/r)*r*f.x
      .= 1*f.x by A2,XCMPLX_1:106;
    hence thesis;
  end;
  dom(r(#)f) = dom f by VALUED_1:def 5;
  then (1/r)(#)(r(#)f)=f by A1,A3,PARTFUN1:5;
  hence thesis by A2,Lm2,XREAL_1:139;
end;
