reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;
reserve C for Function of [:COMPLEX,COMPLEX:],COMPLEX;
reserve G for Function of [:REAL,REAL:],REAL;
reserve h for Function of COMPLEX,COMPLEX,
  g for Function of REAL,REAL;

theorem Th48:
  for x being FinSequence of COMPLEX holds Re (<i>*x) = -(Im x) &
  Im (<i>*x) = Re x
proof
  let x be FinSequence of COMPLEX;
A1: len (x*') = len x by Def1;
A2: Im (<i>*x) =(((-1/2)*<i>))*(<i>*x-((-<i>)*(x*'))) by Th13,COMPLEX1:31
    .=(((-1/2)*<i>))*(<i>*x+-(-(<i>*x*'))) by Th45
    .=(((-1/2)*<i>))*(<i>*(x+x*')) by A1,Th25
    .=(((-1/2)*<i>)*<i>)*(x+x*') by Th44
    .=Re x;
  Re (<i>*x) =(1/2)*(<i>*x+((-<i>)*(x*'))) by Th13,COMPLEX1:31
    .=(1/2)*(<i>*x-(<i>*((x*')))) by Th45
    .=(1/2)*(<i>*(x-(x*'))) by A1,Th36
    .=(1/2*<i>)*(x-x*') by Th44
    .=((-1)*((-1/2)*<i>))*(x-x*')
    .=-(Im x) by Th44;
  hence thesis by A2;
end;
