reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem Th8:
  max(f,min(f,g)) = f & min(f,max(f,g)) = f
proof
A1: C = dom min(f,max(f,g)) by FUNCT_2:def 1;
A2: for x being Element of C st x in C holds max(f,min(f,g)).x = f.x
  proof
    let x be Element of C;
    max(f,min(f,g)).x = max(f.x,min(f,g).x) by Def4
      .= max(f.x,min(f.x,g.x)) by Def3
      .= f.x by XXREAL_0:36;
    hence thesis;
  end;
A3: for x being Element of C st x in C holds min(f,max(f,g)).x = f.x
  proof
    let x be Element of C;
    min(f,max(f,g)).x = min(f.x,max(f,g).x) by Def3
      .= min(f.x,max(f.x,g.x)) by Def4
      .= f.x by XXREAL_0:35;
    hence thesis;
  end;
  C = dom max(f,min(f,g)) & C = dom f by FUNCT_2:def 1;
  hence thesis by A1,A2,A3,PARTFUN1:5;
end;
