reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th41:
  a>0 & b>0 & a|^n + b|^n = c|^n implies
  ex j,k,l st j|^n + k|^n = l|^n & j,k are_coprime
  & j,l are_coprime & k,l are_coprime
  & a = (a gcd b)*j & b = (a gcd b)*k & c = (a gcd b)*l
  proof
    assume
    A1: a>0 & b>0 & a|^n + b|^n = c|^n;
    then n <> 0 by Lm0;
    then consider m such that
    A3: n = 1+m by NAT_1:10,14;
    consider j,k,l such that
    A4: j|^(m+1) + k|^(m+1) = l|^(m+1) & j,k are_coprime
    & a = (a gcd b)*j & b = (a gcd b)*k & c = (a gcd b)*l by A1,A3,Lm58;
    j,l are_coprime & k,l are_coprime by A4,Lm59;
    hence thesis by A3,A4;
  end;
