reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th37:
  |.f.|^ = |. f^ .|
proof
A1: dom ((|.f.|)^) = dom (|.f.|) \ (|.f.|)"{0} by RFUNCT_1:def 2
    .= dom f \ (|.f.|)"{0} by VALUED_1:def 11
    .= dom f \ f"{0c} by Th10
    .= dom (f^) by Def2
    .= dom (|.(f^).|) by VALUED_1:def 11;
  now
    let c;
A2: dom f = dom |.f.| by VALUED_1:def 11;
    assume
A3: c in dom ((|.f.|)^);
    then
A4: c in dom (f^) by A1,VALUED_1:def 11;
    c in dom (|.f.|) \ (|.f.|)"{0} by A3,RFUNCT_1:def 2;
    then
A5: c in dom (|.f.|) by XBOOLE_0:def 5;
    thus ((|.f.|)^).c = ((|.f.|).c)" by A3,RFUNCT_1:def 2
      .= (|.(f.c) .|)" by VALUED_1:18
      .= (|.(f/.c) .|)" by A5,A2,PARTFUN1:def 6
      .= |.1r.|/|.(f/.c) .| by COMPLEX1:48,XCMPLX_1:215
      .= |.1r/(f/.c) .| by COMPLEX1:67
      .= |.((f/.c))" .| by COMPLEX1:def 4,XCMPLX_1:215
      .= |.(f^)/.c .| by A4,Def2
      .= |.(f^).c .| by A4,PARTFUN1:def 6
      .= (|.(f^).|).c by VALUED_1:18;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
