reserve p for Real;
reserve S,T for RealNormSpace;
reserve x0 for Point of S;
reserve f for PartFunc of S,T;
reserve c for constant sequence of S;
reserve R for RestFunc of S,T;
reserve U for RealNormSpace;

theorem Th7:
  for R be RestFunc of S,T st R/.0.S=0.T
  for e be Real st e > 0
  ex d be Real st
  d > 0 & for h be Point of S st ||.h.|| < d holds ||.R/.h.|| <= e*||.h
  .||
proof
  let R be RestFunc of S,T such that
A1: R/.0.S=0.T;
  let e be Real such that
A2: e > 0;
  R is total by NDIFF_1:def 5;
  then consider d be Real such that
A3: d > 0 and
A4: for z be Point of S st z <> 0.S & ||.z.|| < d holds ( ||.z.||"* ||.
  R/.z .||) < e by A2,NDIFF_1:23;
  take d;
  now
    let h be Point of S such that
A5: ||.h.|| < d;
    now
      per cases;
      case
A6:     h <> 0.S;
        then 0 <= ||.h.|| & ||.h.||"*||. R/.h .|| <= e by A4,A5,NORMSP_1:4;
        then ||.h.||*(||.h.||"*||. R/.h .||) <= ||.h.||*e by XREAL_1:64;
        then
A7:     ||.h.||*||.h.||"*||. R/.h .|| <= e* ||.h.||;
        ||.h.|| <> 0 by A6,NORMSP_0:def 5;
        then 1*||. R/.h .|| <= e* ||.h.|| by A7,XCMPLX_0:def 7;
        hence ||. R/.h .|| <= e* ||.h.||;
      end;
      case
A8:     h = 0.S;
        0 <= ||.h.|| by NORMSP_1:4;
        then 0* ||.h.|| <= e* ||.h.|| by A2;
        hence ||. R/.h .|| <= e* ||.h.|| by A1,A8,NORMSP_0:def 6;
      end;
    end;
    hence ||. R/.h .|| <= e* ||.h.||;
  end;
  hence thesis by A3;
end;
