reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;

theorem Th21:
  for f being (the carrier of L)-valued Function
  for a being object holds
  Support(f+*(a,x)) c= Support f \/ {a}
  proof
    let f be (the carrier of L)-valued Function;
    let a,z be object;
    set g = f+*(a,x);
    assume
A1: z in Support(g);
    a = z or z in Support f
    proof
      assume a <> z;
      then
A2:   g.z = f.z by FUNCT_7:32;
      dom g = dom f by FUNCT_7:30;
      then z in dom f & g.z <> 0.L by A1,POLYNOM1:def 3;
      hence thesis by A2,POLYNOM1:def 3;
    end;
    hence thesis by ZFMISC_1:136;
  end;
