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 Th50:
  for x,y being FinSequence of COMPLEX st len x=len y holds |(x,
  <i>*y)| = -<i>*(|(x,y)|)
proof
  let x,y be FinSequence of COMPLEX;
  assume
A1: len x=len y;
A2: len (Im x) = len x by Th40;
A3: len (Re x) = len x by Th40;
A4: len (Im y) = len y by Th40;
  |(x,<i>*y)| =|(Re x,-(Im y))| - <i>*(|(Re x,Im (<i>*y))|) + <i>*(|(Im x,
  Re (<i>*y))|) + |(Im x,Im (<i>*y))| by Th48
    .=|(Re x,-(Im y))| - <i>*(|(Re x,Re y)|) + <i>*(|(Im x,Re (<i>*y))|) +
  |(Im x,Im (<i>*y))| by Th48
    .=|(Re x,-(Im y))| - <i>*(|(Re x,Re y)|) + <i>*(|(Im x,-Im y)|) + |(Im x
  ,Im (<i>*y))| by Th48
    .=|(Re x,-(Im y))| - <i>*(|(Re x,Re y)|) + <i>*(|(Im x,-Im y)|) + |(Im x
  ,Re y)| by Th48
    .=-|(Re x,Im y)| - <i>*(|(Re x,Re y)|) + <i>*(|(Im x,-Im y)|) + |(Im x,
  Re y)| by A1,A3,A4,RVSUM_1:122
    .=-|(Re x,Im y)| - <i>*(|(Re x,Re y)|) + <i>*(-(|(Im x,Im y)|)) + |(Im x
  ,Re y)| by A1,A2,A4,RVSUM_1:122
    .=-<i>*(|(x,y)|);
  hence thesis;
end;
