reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th7:
for f,g be PartFunc of REAL i,REAL
 holds <>*(f+g) = <>*f + <>*g & <>*(f-g) = <>*f - <>*g
proof
   let f,g be PartFunc of REAL i,REAL;
A1:dom (<>*(f+g)) = dom (f+g) & dom (<>*(f-g)) = dom (f-g) &
   dom (<>*f) = dom f & dom (<>*g) = dom g by Th3; then
   dom (<>*(f+g)) = dom (<>*f) /\ dom (<>*g) &
   dom (<>*(f-g)) = dom (<>*f) /\ dom (<>*g) by VALUED_1:12,def 1; then
A2:dom (<>*(f+g)) = dom (<>*f+<>*g) &
   dom (<>*(f-g)) = dom (<>*f-<>*g) by VALUED_2:def 45,def 46;
   now let x be object;
    assume A3:x in dom (<>*(f+g)); then
    x in dom f /\ dom g by A1,VALUED_1:def 1; then
    x in dom f & x in dom g by XBOOLE_0:def 4; then
A4: <* f.x *> = (<>*f).x & <* g.x *> = (<>*g).x by Th6;
    (<>*(f+g)).x = <* (f+g).x *> by Th6,A3,A1
                .= <* f.x+g.x *> by A3,A1,VALUED_1:def 1
                .= (<>*f).x + (<>*g).x by A4,RVSUM_1:13;
    hence (<>*(f+g)).x = (<>*f+ <>*g).x by A2,A3,VALUED_2:def 45;
   end;
   hence <>*(f+g) = <>*f + <>*g by A2,FUNCT_1:2;
   now let x be object;
    assume A5:x in dom (<>*(f-g)); then
    x in dom f /\ dom g by A1,VALUED_1:12; then
    x in dom f & x in dom g by XBOOLE_0:def 4; then
A6: <* f.x *> = (<>*f).x & <* g.x *> = (<>*g).x by Th6;
    thus (<>*(f-g)).x = <* (f-g).x *> by Th6,A5,A1
                .= <* f.x-g.x *> by A5,A1,VALUED_1:13
                .= (<>*f).x - (<>*g).x by A6,RVSUM_1:29
                .= (<>*f- <>*g).x by A2,A5,VALUED_2:def 46;
   end;
   hence thesis by A2,FUNCT_1:2;
end;
