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

theorem
for f,g be PartFunc of REAL m,REAL
  st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g
  holds f+g is_continuous_on Z & f-g is_continuous_on Z
proof
   let f,g be PartFunc of REAL m,REAL;
   assume f is_continuous_on Z & g is_continuous_on Z &
   Z c= dom f & Z c= dom g;
   then
   <>*f is_continuous_on Z & <>*g is_continuous_on Z by Th44; then
A1:<>*f + <>*g is_continuous_on Z & <>*f - <>*g is_continuous_on Z by Th33;
   <>*f + <>*g = <>*(f+g) & <>*f - <>*g = <>*(f-g) by Th7;
   hence thesis by A1,Th44;
end;
