reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th41:
  f is_Lipschitzian_on X implies p(#)f is_Lipschitzian_on X
proof
  assume
A1: f is_Lipschitzian_on X;
  then consider s such that
A2: 0<s and
A3: for x1,x2 st x1 in X & x2 in X holds ||. f/.x1-f/.x2.||<=s*||. x1-x2
  .||;
  X c= dom f by A1;
  hence
A4: X c= dom (p(#)f) by VFUNCT_1:def 4;
  per cases;
  suppose
A5: p=0;
    take s;
    thus 0<s by A2;
    let x1,x2;
    assume that
A6: x1 in X and
A7: x2 in X;
    0<=||. x1-x2.|| by NORMSP_1:4;
    then
A8: s*0<=s*||. x1-x2.|| by A2;
    ||. (p(#)f)/.x1-(p(#)f)/.x2.|| = ||. p*(f/.x1)-(p(#)f)/.x2.||
    by A4,A6,VFUNCT_1: def 4
    .= ||. 0.T-(p(#)f)/.x2.|| by A5,RLVECT_1:10
    .= ||. 0.T - p*(f/.x2).|| by A4,A7,VFUNCT_1:def 4
    .= ||. 0.T-0.T.|| by A5,RLVECT_1:10
    .= ||. 0.T.||
    .= 0 by NORMSP_1:1;
    hence ||. (p(#)f)/.x1-(p(#)f)/.x2.|| <=s*||. x1-x2.|| by A8;
  end;
  suppose
A9: p<>0;
    take g = |.p.| *s;
    0<|.p.| by A9,COMPLEX1:47;
    then 0*s<|.p.|*s by A2,XREAL_1:68;
    hence 0<g;
    let x1,x2;
    assume that
A10: x1 in X and
A11: x2 in X;
     0<=|.p.| by COMPLEX1:46;
     then
A12: |.p.|*||. f/.x1-f/.x2.|| <=|.p.|*(s*||. x1-x2.||)
     by A3,A10,A11,XREAL_1:64;
     ||. (p(#)f)/.x1-(p(#)f)/.x2.|| = ||. p*(f/.x1)-(p(#)f)/.x2.||
     by A4,A10,VFUNCT_1: def 4
     .= ||. p*(f/.x1) - p*(f/.x2).|| by A4,A11,VFUNCT_1:def 4
     .= ||. p*(f/.x1 - f/.x2).|| by RLVECT_1:34
     .= |.p.|*||. f/.x1 - f/.x2.|| by NORMSP_1:def 1;
     hence ||. (p(#)f)/.x1-(p(#)f)/.x2.|| <= g*||. x1-x2.|| by A12;
   end;
 end;
