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 Th49:
  for x,y being FinSequence of COMPLEX st len x=len y holds
  |(<i>*x,y)| = <i>*(|(x,y)|)
proof
  let x,y be FinSequence of COMPLEX;
  assume
A1: len x=len y;
A2: len (Im y) = len y by Th40;
A3: len (Re y) = len y by Th40;
A4: len (Im x) = len x by Th40;
  |(<i>*x,y)| =|(-(Im x),Re y)| - <i>*(|(Re (<i>*x),Im y)|) + <i>*(|(Im (
  <i>*x),Re y)|) + |(Im (<i>*x),Im y)| by Th48
    .=|(-(Im x),Re y)| - <i>*(|(-(Im x),Im y)|) + <i>*(|(Im (<i>*x),Re y)|)
  + |(Im (<i>*x),Im y)| by Th48
    .=|(-(Im x),Re y)| - <i>*(|(-(Im x),Im y)|) + <i>*(|(Re x,Re y)|) + |(Im
  (<i>*x),Im y)| by Th48
    .=|(-(Im x),Re y)| - <i>*(|(-(Im x),Im y)|) + <i>*(|(Re x,Re y)|) + |(Re
  x,Im y)| by Th48
    .=-|((Im x),Re y)| - <i>*(|(-(Im x),Im y)|) + <i>*(|(Re x,Re y)|) + |(Re
  x,Im y)| by A1,A3,A4,RVSUM_1:122
    .=-|((Im x),Re y)| - <i>*(-|((Im x),Im y)|) + <i>*(|(Re x,Re y)|) + |(Re
  x,Im y)| by A1,A4,A2,RVSUM_1:122
    .=<i>*(|(x,y)|);
  hence thesis;
end;
