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

theorem Th9:
for f be PartFunc of REAL i,REAL, g be PartFunc of REAL i,REAL 1
 st <>*f = g holds |.f.| = |.g.|
proof
   let f be PartFunc of REAL i,REAL, g be PartFunc of REAL i,REAL 1;
   assume A1: <>*f = g;
A2:dom |.g.| = dom g by NFCONT_4:def 2
            .= dom f by A1,Th3; then
A3:dom |.g.| = dom |.f.| by VALUED_1:def 11;
   now let x be Element of REAL i;
    assume A4: x in dom |.g.|; then
A5: g/.x = <* f/.x *> by A1,A2,Th6;
    thus (|.g.|).x = (|.g.|)/.x by A4,PARTFUN1:def 6
      .= |. g/.x .| by A4,NFCONT_4:def 2
      .= |. f/.x .| by A5,Lm1
      .= |. f.x .| by A2,A4,PARTFUN1:def 6
      .= (|.f.|).x by VALUED_1:18;
   end;
   hence thesis by A3,PARTFUN1:5;
end;
