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;

theorem Th41:
  for x,y being FinSequence of COMPLEX st len x=len y holds Re (x+
  y) = Re x + Re y & Im(x + y) = Im x + Im y
proof
  let x,y be FinSequence of COMPLEX;
A1: len (-x*')=len (x*') by Th5;
  assume
A2: len x=len y;
  then
A3: len (x+y)=len x by Th6;
A4: len y=len (y*') by Def1;
  then
A5: len (y+(y*'))=len y by Th6;
A6: len x=len (x*') by Def1;
  then
A7: len (x+(x*'))=len x by Th6;
A8: len (x-(x*'))=len x by A6,Th7;
A9: len (y-(y*'))=len y by A4,Th7;
  thus Re (x+y)=1/2*((x+y)+(x*'+y*')) by A2,Th15
    .=1/2*((x+y+x*')+y*') by A2,A4,A6,A3,Th24
    .=1/2*((x+x*'+y)+y*') by A2,A6,Th24
    .=1/2*((x+x*')+(y+y*')) by A2,A4,A7,Th24
    .= Re x + Re y by A2,A7,A5,Th25;
  thus Im (x+y)=(-1/2*<i>)*((x+y)-(x*'+y*')) by A2,Th15
    .=(-1/2*<i>)*((x+y)-x*'-y*') by A2,A4,A6,A3,Th30
    .=(-1/2*<i>)*(x+-x*'+y-y*') by A2,A6,A1,Th24
    .=(-1/2*<i>)*((x-x*')+(y-y*')) by A2,A4,A8,Th31
    .=Im x + Im y by A2,A8,A9,Th25;
end;
