
theorem Th2:
for X be non empty set, f be PartFunc of X,ExtREAL st f is real-valued
 holds -f is real-valued & max+f is real-valued & max-f is real-valued
proof
    let X be non empty set, f be PartFunc of X,ExtREAL;
    assume f is real-valued; then
A1: rng f c= REAL by VALUED_0:def 3;
    now let y be object;
     assume y in rng(-f); then
     consider x be Element of X such that
A2:   x in dom(-f) & y = (-f).x by PARTFUN1:3;
     x in dom f by A2,MESFUNC1:def 7; then
A3:  f.x in REAL by A1,FUNCT_1:3;
     y = -(f.x) by A2,MESFUNC1:def 7;
     hence y in REAL by A3,XREAL_0:def 1;
    end; then
A4: rng(-f) c= REAL;
    hence -f is real-valued by VALUED_0:def 3;
A5: rng f \/ {0} c= REAL by A1,XBOOLE_1:8;
    rng max+f c= rng f \/ {0} by Th1; then
    rng max+f c= REAL by A5;
    hence max+f is real-valued by VALUED_0:def 3;
A6: rng(-f) \/ {0} c= REAL by A4,XBOOLE_1:8;
    rng max-f c= rng(-f) \/ {0} by Th1; then
    rng max-f c= REAL by A6;
    hence max-f is real-valued by VALUED_0:def 3;
end;
