
theorem Th59:
for f be PartFunc of REAL,REAL, A be non empty closed_interval Subset of REAL
 st A c= dom f holds max+(f||A) = max+(f|A) & max-(f||A) = max-(f|A)
proof
    let f be PartFunc of REAL,REAL,
     A be non empty closed_interval Subset of REAL;
     assume A c= dom f;
A1:  dom(max+(f||A)) = dom(f|A) & dom(f|A) = dom(max+(f|A))
       by RFUNCT_3:def 10;
     for x be object st x in dom(max+(f||A)) holds
      (max+(f||A)).x = (max+(f|A)).x
     proof
      let x be object;
      assume
A2:    x in dom(max+(f||A)); then
      (max+(f||A)).x = max+((f|A).x) by RFUNCT_3:def 10;
      hence (max+(f||A)).x = (max+(f|A)).x by A2,A1,RFUNCT_3:def 10;
     end;
     hence max+(f||A) = max+(f|A) by A1,FUNCT_1:2;

A3:  dom(max-(f||A)) = dom(f|A) & dom(f|A) = dom(max-(f|A))
       by RFUNCT_3:def 11;
     for x be object st x in dom(max-(f||A)) holds
      (max-(f||A)).x = (max-(f|A)).x
     proof
      let x be object;
      assume
A4:    x in dom(max-(f||A)); then
      (max-(f||A)).x = max-((f|A).x) by RFUNCT_3:def 11;
      hence (max-(f||A)).x = (max-(f|A)).x by A4,A3,RFUNCT_3:def 11;
     end;
     hence max-(f||A) = max-(f|A) by A3,FUNCT_1:2;
end;
