
theorem
  for X be RealUnitarySpace, w be Point of X,
      f be Function of X,REAL
  st for v be Point of X holds f.v = w .|. v
    holds f is_continuous_on the carrier of X
proof
  let X be RealUnitarySpace, w be Point of X,
      f be Function of X, REAL;
  assume AS: for v be Point of X holds f.v = w .|. v;
  set Y=RUSp2RNSp X;
  reconsider g=f as Function of Y,REAL;
A3: dom g = the carrier of Y by FUNCT_2:def 1;
  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;
    for r be Real st 0 < r ex s be Real st
      0 < s &
      for y1 be Point of Y st y1 in dom g & ||. y1- y0 .|| < s
        holds |. g/.y1 - g/.y0 .| < r
    proof
      let r be Real;
      assume AS1: 0 < r;
      thus ex s be Real st 0 < s &
        for y1 be Point of Y st y1 in dom g & ||. y1 - y0 .|| < s
          holds |. g/.y1 - g/.y0 .| < r
      proof
C1:     0 <= ||.w.|| by BHSP_1:28;
        reconsider s=r/(||.w.|| + 1) as Real;
C41:    ||.w.|| + 0 < ||.w.|| + 1 by XREAL_1:8;
        s*(||.w.|| + 1) = r by C1,XCMPLX_1:87; then
C5:     0 < s & s*||.w.|| < r by C1,AS1,C41,XREAL_1:68;
C6:     for y1 be Point of Y st y1 in dom g & ||. y1 - y0 .|| < s
          holds |. g/.y1 - g/.y0 .| < r
        proof
          let y1 be Point of Y;
          assume AS2: y1 in dom g & ||. y1 - y0 .|| < s;
          reconsider x1=y1 as Point of X;
          reconsider x0=y0 as Point of X;
X0:       ||. y1 - y0 .|| = ||. x1 - x0 .|| by RHS4,RHS6;
D1:       |. g/.y1 - g/.y0 .| = |. w .|. x1 - g.y0 .| by AS
                             .= |. w .|. x1 - w .|. x0 .| by AS
                             .= |. w .|. (x1 - x0) .| by BHSP_1:12;
D2:       |. w .|. (x1 - x0) .| <= ||.w.||*||.x1 - x0.|| by BHSP_1:29;
          ||.w.||*||.x1 - x0.|| <= ||.w.||*s by X0,C1,AS2,XREAL_1:64; then
          |. g/.y1 - g/.y0 .| <= ||.w.||*s by D1,D2,XXREAL_0:2;
          hence |. g/.y1 - g/.y0 .| < r by C5,XXREAL_0:2;
        end;
        take s;
        thus thesis by C1,AS1,C6;
      end;
    end;
    hence g| (the carrier of Y) is_continuous_in y0 by A3,NFCONT_1:8;
  end; then
  g is_continuous_on the carrier of Y by FUNCT_2:def 1;
  hence thesis by LM3C;
end;
