
theorem LM3C:
  for X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL
  st f=g holds
   f is_continuous_on the carrier of X
 iff
   g is_continuous_on the carrier of RUSp2RNSp X
proof
  let X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL;
  assume AS: f=g;
  set Y=RUSp2RNSp X;
  hereby assume A11: f is_continuous_on the carrier of X;
    for y0 be Point of Y st y0 in the carrier of Y
      holds g| (the carrier of Y) is_continuous_in y0
    proof
      let y0 be Point of Y;
      assume y0 in the carrier of Y;
      reconsider x0=y0 as Point of X;
      f| (the carrier of X) is_continuous_in x0 by A11;
      hence thesis by LM3B,AS;
    end;
    hence g is_continuous_on the carrier of Y by A11,AS;
  end;
  assume A31: g is_continuous_on the carrier of Y;
  for x0 be Point of X st x0 in the carrier of X
      holds f| (the carrier of X) is_continuous_in x0
  proof
    let x0 be Point of X;
    assume x0 in the carrier of X;
    reconsider y0=x0 as Point of Y;
    g| (the carrier of Y) is_continuous_in y0 by A31;
    hence thesis by LM3B,AS;
  end;
  hence thesis by A31,AS;
end;
