reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th59:
  for F be PartFunc of REAL,REAL, X be set, r st F
  is_convex_on X holds max+(F-r) is_convex_on X
proof
  let F be PartFunc of REAL,REAL, X be set, r;
  assume
A1: F is_convex_on X;
  then
A2: X c= dom F;
A3: dom F = dom(F-r) & dom(max+(F-r)) = dom(F-r) by Def10,VALUED_1:3;
  hence X c= dom(max+(F-r)) by A1;
  let p be Real;
  assume that
A4: 0<=p and
A5: p<=1;
  let x,y be Real;
  assume that
A6: x in X and
A7: y in X and
A8: p*x+(1-p)*y in X;
  F.(p*x+(1-p)*y) <= p*F.x + (1-p)*F.y by A1,A4,A5,A6,A7,A8;
  then F.(p*x+(1-p)*y) -r <= p*F.x + (1-p)*F.y -r by XREAL_1:9;
  then
A9: max+(F.(p*x+(1-p)*y) -r) <= max(p*F.x + (1-p)*F.y -r,0) by XXREAL_0:26;
  0+p<=1 by A5;
  then 0<=1-p by XREAL_1:19;
  then
A10: max+((1-p)*(F-r).y) = (1-p)*(max+ ((F-r).y)) by Th4;
A11: max+(p*(F-r).x + (1-p)*(F-r).y)<= max+(p*(F-r).x) + max+((1-p)*(F-r).y)
  by Th5;
A12: max+(p*(F-r).x)= p* (max+ ((F-r).x)) by A4,Th4;
  reconsider pc = p*x+(1-p)*y as Element of REAL by XREAL_0:def 1;
  reconsider x,y as Element of REAL by XREAL_0:def 1;
  p*F.x + (1-p)*F.y - r = p*(F.x -r) + (1-p)*(F.y - r)
    .= p*(F-r).x + (1-p)*(F.y - r) by A6,A2,VALUED_1:3
    .= p*(F-r).x + (1-p)*(F-r).y by A7,A2,VALUED_1:3;
  then
  max+(F.(p*x+(1-p)*y) -r) <= p*(max+ ((F-r).x)) + (1-p)*(max+ ((F-r).y))
  by A9,A11,A12,A10,XXREAL_0:2;
  then
  max+ ((F-r).pc) <= p*(max+ ((F-r).x)) + (1-p)*(max+ ((F-r).y
  )) by A8,A2,VALUED_1:3;
  then
  (max+ (F-r)).pc <= p*(max+ ((F-r).x)) + (1-p)*(max+ ((F-r).y
  )) by A3,A8,A2,Def10;
  then
  (max+ (F-r)).pc <= p*(max+ (F-r)).x + (1-p)*(max+ ((F-r).y))
  by A3,A6,A2,Def10;
  hence thesis by A3,A7,A2,Def10;
end;
