reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th5:
for f be PartFunc of REAL i,REAL holds <>*(f|Z) = (<>*f) |Z
proof
   let f be PartFunc of REAL i,REAL;
   set W = proj(1,1) qua Function";
   rng (f|Z) c= dom W by PDIFF_1:2; then
   dom(W*(f|Z)) = dom (f|Z) by RELAT_1:27
               .= dom f /\ Z by RELAT_1:61; then
A1:dom(W*(f|Z)) = (dom <>*f) /\ Z by Th3;
   now let x be object;
    assume A2: x in dom ((<>*f) | Z); then
    x in (dom <>*f) /\ Z by RELAT_1:61; then
    x in dom f /\ Z by Th3; then
A3: x in Z & x in dom f by XBOOLE_0:def 4;
    dom(W*(f|Z)) = dom ((<>*f) | Z) by A1,RELAT_1:61; then
    (<>*(f|Z)).x = W.((f|Z).x) by A2,FUNCT_1:12
                .= W.(f.x) by A3,FUNCT_1:49
                .= (W*f).x by A3,FUNCT_1:13;
    hence (<>*(f|Z)).x =((<>*f) |Z).x by A2,FUNCT_1:47;
   end;
   hence thesis by A1,FUNCT_1:2,RELAT_1:61;
end;
