reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem Th2:
  for c,z being Complex holds c + <*z*> = <*c+z*>
  proof
    let c,z be Complex;
A1: len <*z*> = 1 by FINSEQ_1:39;
A2: len <*c+z*> = 1 by FINSEQ_1:39;
A3: len(c+<*z*>) = len <*z*> by CARD_1:def 7;
    thus len(c+<*z*>) = len <*c+z*> by A1,A2,CARD_1:def 7;
    let k such that
A4: 1 <= k & k <= len(c+<*z*>);
A5: k = 1 by A1,A3,A4,XXREAL_0:1;
    k in dom(c+<*z*>) by A4,FINSEQ_3:25;
    hence (c+<*z*>).k = c+(<*z*>.k) by VALUED_1:def 2
    .= <*c+z*>.k by A5;
  end;
