reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;

theorem Th1918:
  for f be PartFunc of REAL,the carrier of Z st
    A c= dom f holds ||.f|A.|| = (||.f.||) |A
proof
   let f be PartFunc of REAL,the carrier of Z;
   assume A1: A c= dom f; then
A2:dom (f|A) = A by RELAT_1:62;
A3:dom ((||.f.||) |A) = dom ||.f.|| /\ A by RELAT_1:61; then
A6:dom ((||.f.||) |A) = dom f /\ A by NORMSP_0:def 3; then
   dom ((||.f.||) |A) = A by A1,XBOOLE_1:28; then
A4:dom ||. f|A .|| = dom ((||.f.||) |A) by A2,NORMSP_0:def 3;
   now let x be object;
    assume A5: x in dom ((||.f.||) |A); then
A9: x in dom (||.f.||) & x in dom f by A3,A6,XBOOLE_0:def 4; then
A11:f/.x = f.x by PARTFUN1:def 6
      .= (f|A).x by FUNCT_1:49,A5
      .= (f|A)/.x by A2,PARTFUN1:def 6,A5;
    thus ((||.f.||) |A).x = (||.f.||).x by FUNCT_1:49,A5
      .= ||. f/.x .|| by A9,NORMSP_0:def 3
      .= (||. (f|A) .||) .x by A11,A4,A5,NORMSP_0:def 3;
   end;
   hence thesis by A4,FUNCT_1:2;
end;
