theorem Th64:
  for x,y being FinSequence of COMPLEX holds |(x,y)| = (|(y,x)|)*'
proof
  let x,y be FinSequence of COMPLEX;
  set x1=|(Re y,Re x)|,x2=|(Re y,Im x)|,y1=|(Im y,Re x)|, y2=|(Im y,Im x)|;
  reconsider x19=x1, x29=x2, y19=y1, y29=y2 as Element of REAL
    by XREAL_0:def 1;
A1: Im x19=0 by COMPLEX1:def 2;
A2: Im x29=0 by COMPLEX1:def 2;
A3: Im y29=0 by COMPLEX1:def 2;
A4: Im y19=0 by COMPLEX1:def 2;
  reconsider x19=x1, x29=x2, y19=y1, y29=y2 as Element of COMPLEX by
XCMPLX_0:def 2;
  (|(y,x)|)*' =(x19 + y29 + <i>*((y19) - (x29)))*'
    .=(x19 + y29)*' + (<i>*((y19) - (x29)))*' by COMPLEX1:32
    .=(x19)*' + (y29)*' + (<i>*((y19) - (x29)))*' by COMPLEX1:32
    .=x19 + (y29)*' + (<i>*((y19) - (x29)))*' by A1,COMPLEX1:38
    .=x19 + y29 + (<i>*((y19) - (x29)))*' by A3,COMPLEX1:38
    .=x19 + y29 + (-<i>)*((y19) - (x29))*' by COMPLEX1:31,35
    .=x19 + y29 + (-<i>)*((y19)*' - (x29)*') by COMPLEX1:34
    .=x19 + y29 + (-<i>)*((y19) - (x29)*') by A4,COMPLEX1:38
    .=x19 + y29 + (-<i>)*((y19) - (x29)) by A2,COMPLEX1:38
    .=|(x, y)|;
  hence thesis;
end;
