
theorem LM3B:
  for X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL,
      x0 be Point of X, y0 be Point of RUSp2RNSp X
  st f=g & x0=y0 holds
   f is_continuous_in x0 iff g is_continuous_in y0
proof
  let X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL,
      x0 be Point of X, y0 be Point of RUSp2RNSp X;
  assume AS: f=g & x0=y0;
  set Y=RUSp2RNSp X;
  hereby assume A11: f is_continuous_in x0;
    for s2 be sequence of Y
      st rng s2 c= dom g & s2 is convergent & lim s2 = y0
     holds g/*s2 is convergent & g/.y0 = lim (g/*s2)
    proof
      let s2 be sequence of Y;
      assume AS1: rng s2 c= dom g & s2 is convergent & lim s2 = y0;
      reconsider s1=s2 as sequence of X;
B2:   s1 is convergent by AS1,RHS8; then
      lim s1 = x0 by AS1,AS,RHS9;
      hence g/*s2 is convergent & g/.y0 = lim (g/*s2) by AS,A11,AS1,B2;
    end;
    hence g is_continuous_in y0 by A11,AS;
  end;
  assume A31: g is_continuous_in y0;
  for s1 be sequence of X
    st rng s1 c= dom f & s1 is convergent & lim s1 = x0
   holds f/*s1 is convergent & f/.x0 = lim (f/*s1)
  proof
    let s1 be sequence of X;
    assume AS2: rng s1 c= dom f & s1 is convergent & lim s1 = x0;
    reconsider s2=s1 as sequence of Y;
B2: s2 is convergent by AS2,RHS8;
    lim s2 = y0 by AS,AS2,RHS9;
    hence f/*s1 is convergent & f/.x0 = lim (f/*s1) by AS,A31,AS2,B2;
  end;
  hence f is_continuous_in x0 by A31,AS;
end;
