
theorem Th60:
for X be non empty set, f,g be PartFunc of X,ExtREAL
 holds -(f+g) = (-f) + (-g) & -(f-g) = (-f) + g
     & -(f-g) = g - f & -(-f+g) = f - g & -(-f+g) = f + (-g)
proof
   let X be non empty set, f,g be PartFunc of X,ExtREAL;
A1:f"{-infty} = (-f)"{+infty} & f"{+infty} = (-f)"{-infty} &
   g"{-infty} = (-g)"{+infty} & g"{+infty} = (-g)"{-infty} by LEM10;
A2:dom f = dom(-f) & dom g = dom (-g) by MESFUNC1:def 7;

A3:dom(-(f+g)) = dom(f+g) & dom(-f) = dom f & dom(-g) = dom g
 & dom(-(f-g)) = dom(f-g) by MESFUNC1:def 7; then
A4:dom(-(f+g)) = (dom f /\ dom g) \ ((f"{-infty} /\ g"{+infty}) \/
     (f"{+infty} /\ g"{-infty})) by MESFUNC1:def 3; then
A5:dom(-(f + g)) = dom((-f) + (-g)) by A1,A2,MESFUNC1:def 3;
A6:dom(-(f-g)) = (dom f /\ dom g) \ ((f"{+infty} /\ g"{+infty}) \/
     (f"{-infty} /\ g"{-infty})) by A3,MESFUNC1:def 4; then
A7:dom(-(f-g)) = dom((-f)+g) by A1,A2,MESFUNC1:def 3; then
C1:dom(-(-f+g)) = dom(-(f-g)) by MESFUNC1:def 7; then
A10:dom(-(-f+g)) = dom(f-g) by MESFUNC1:def 7; then
   dom(-(-f+g)) = (dom f /\ dom g) \ ((f"{+infty} /\ g"{+infty}) \/
      (f"{-infty} /\ g"{-infty})) by MESFUNC1:def 4; then
A12:dom(-(-f+g)) = dom(f+(-g)) by A1,A2,MESFUNC1:def 3;
A8:dom(-(f-g)) = dom(g-f) by A6,MESFUNC1:def 4;
A9:dom(-(-f+g)) = dom(-f+g) by MESFUNC1:def 7;
B3:dom(-(f+g)) c= dom f & dom(-(f+g)) c= dom g by A4,XBOOLE_1:18,36;
B4:dom(-(f-g)) c= dom f & dom(-(f-g)) c= dom g by A6,XBOOLE_1:18,36;
B5:dom(-(-f+g)) c= dom (-f) & dom(-(-f+g)) c= dom g
     by C1,A3,A6,XBOOLE_1:18,36;
   now let x be Element of X;
    assume B2: x in dom(-(f+g)); then
    (-(f+g)).x = -((f+g).x) by MESFUNC1:def 7
      .= -(f.x + g.x) by A3,B2,MESFUNC1:def 3
      .= -(f.x) + -(g.x) by XXREAL_3:9
      .= (-f).x + -(g.x) by A2,B2,B3,MESFUNC1:def 7
      .= (-f).x + (-g).x by A2,B2,B3,MESFUNC1:def 7;
    hence (-(f+g)).x = ((-f)+(-g)).x by B2,A5,MESFUNC1:def 3;
   end;
   hence -(f + g) = (-f) + (-g) by A5,PARTFUN1:5;
   now let x be Element of X;
    assume B2: x in dom(-(f-g)); then
    (-(f-g)).x = -((f-g).x) by MESFUNC1:def 7
     .= -(f.x - g.x) by A3,B2,MESFUNC1:def 4
     .= -(f.x) + g.x by XXREAL_3:26
     .= (-f).x + g.x by A2,B4,B2,MESFUNC1:def 7;
    hence (-(f-g)).x = ((-f)+g).x by B2,A7,MESFUNC1:def 3;
   end;
   hence -(f - g) = (-f) + g by A7,PARTFUN1:5;
   now let x be Element of X;
    assume B2: x in dom(-(f-g)); then
    (-(f-g)).x = -((f-g).x) by MESFUNC1:def 7
     .= -(f.x - g.x) by A3,B2,MESFUNC1:def 4
     .= g.x - f.x by XXREAL_3:26;
    hence (-(f-g)).x = (g-f).x by B2,A8,MESFUNC1:def 4;
   end;
   hence -(f-g) = g-f by A8,PARTFUN1:5;
   now let x be Element of X;
    assume B2: x in dom(-(-f+g)); then
    (-(-f+g)).x = -((-f+g).x) by MESFUNC1:def 7
     .= -( (-f).x + g.x ) by A9,B2,MESFUNC1:def 3
     .= -( -(f.x) + g.x ) by B5,B2,MESFUNC1:def 7
     .= f.x - g.x by XXREAL_3:27;
    hence (-(-f+g)).x = (f-g).x by B2,A10,MESFUNC1:def 4;
   end;
   hence -(-f+g) = f-g by A10,PARTFUN1:5;
   now let x be Element of X;
    assume B2: x in dom(-(-f+g)); then
    (-(-f+g)).x = -((-f+g).x) by MESFUNC1:def 7
     .= -( (-f).x + g.x ) by A9,B2,MESFUNC1:def 3
     .= -( -(f.x) + g.x ) by B5,B2,MESFUNC1:def 7
     .= (f.x) + -(g.x) by XXREAL_3:27
     .= (f.x) + (-g).x by B2,B5,A3,MESFUNC1:def 7;
    hence (-(-f+g)).x = (f+(-g)).x by B2,A12,MESFUNC1:def 3;
   end;
   hence -(-f+g) = f + (-g) by A12,PARTFUN1:5;
end;
